Technical Committee 105 - Kasetsart University

Technical Committee 105Geomechanics from Micro to Macro

Comité technique 105Géomécanique micro-macro

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General Report of TC 105 Geomechanics through the scales :

Rapport général du TC 105 La géomécanique à travers les échelles

Viggiani G. UJF-Grenoble / Grenoble-INP / CNRS UMR 5521, Laboratoire 3SR, Grenoble, France

ABSTRACT: This general report presents and discusses the papers submitted to the Discussion Session of TC105 (Geomechanics from Micro to Macro) at the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013. These papers deal with a variety of issues and include experimental, analytical and numerical studies. Overall, they show that the theme of this session is both challenging and promising. The discussion of the papers is preceded by some general remarks about the meaning,trends and perspectives of research in geomechanics through the scales.

RÉSUMÉ : Ce rapport général passe en revue les articles soumis à la 18ème Conférence Internationale de Paris dans le cadre de la session consacrée à la géomécanique de la micro à la macro échelle. Ces articles portent sur plusieurs thématiques, et comprennentdes études soit expérimentales, soit analytiques, soit numériques. Dans leur ensemble, ils montrent bien que le thème de cette sessionest à la fois complexe et riche de perspectives. La discussion des articles est précédée par quelques considérations générales sur le sens et les perspectives actuelles de la recherche dans ce domaine.

KEYWORDS: geomechanics through the scales, experiments, analytical and numerical studies.

1 INTRODUCTION

This general report briefly presents the 15 papers that were submitted to the Discussion Session of TC105 (Geomechanics from Micro to Macro) at the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013. While only 6 papers have been actually selected for oral presentation, all 15 can be found in these proceedings. The focus of this session is on new advances in geomechanics through the scales – from micro to macro – and it encompasses many aspects of geomechanics and geotechnical engineering, from fundamental modeling and laboratory experimental studies to more applied topics and recent challenges related to the safeguard of the environment and the production of energy.

Table 1 lists the papers belonging to the session. In the report, the papers will be presented and discussed in the order adopted in Table 1. Citations of papers belonging to this session will be mentioned in Italics in the text. These papers might be grouped into different tracks, depending on whether they are categorized by material studied (fine-grained vs. coarse-grained geomaterials), approach of the study (experimental, numerical and analytical) or application (petroleum, energy, civil …). In this report, we have rather decided to discuss papers depending on whether they are actually dealing with both the micro and the macro scale, or just focusing either on the micro or the macro scale. The discussion of the papers is preceded by a short summary of the activities of TC105, followed by some general comments about the meaning, trends and perspectives of research in geomechanics through the scales.

2 GEOMECHANICS FROM MICRO TO MACRO

2.1 A short summary of the activities of TC105

At the 15th ICSMGE in Istanbul (2001) a technical committee was appointed (at that time TC35, now TC105) to promote scientific research on the behavior of geomaterials at the micro scale, so as to clarify the fundamental micromechanisms

responsible for behavior observed at higher scales. Since then, the study of the behavior of geomaterials (soils and rocks) through different scales – from micro to macro – has become an emerging field in our community, along with the increasing awareness of the need of integrating these different scales – which is increasingly possible thanks to major advances in both the experimental and computational tools available.

Three specific conferences have been organized by the technical committee TC105 (IS-Yamaguchi in 2006, IS-Shanghai in 2010 and IS-Hong Kong in 2013), and a fourth conference will take place in Cambridge in 2014. Furthermore, a themed issue for Géotechnique (entitled “Soil mechanics at the grain scale”) appeared in 2010, followed by a themed issue for Géotechnique Letters (entitled “Geomechanics across the scales”) in 2012.

2.2 General considerations

Geomaterials are rich in features interacting across the scales – from asperity size to grain size, from the length of force chains to the thickness of shear bands, and from laboratory specimens to the full geotechnical engineering scale. Geomaterials exhibit multi-scale behavior that is intrinsically associated with the interactions of the individual particles (see for example the magnificent review paper by Santamarina 2003). Large-scale geotechnical engineering could gain so much from accurate description of the relevant features exhibited at the finer scales.

The “father” of our discipline, Karl Terzaghi, already reflected on this matter back in 1920: ‘[Coulomb] purposely ignored the fact that sand consists of individual grains. Coulomb’s idea proved very useful as a working hypothesis, but it developed into an obstacle against further progress as soon as its hypothetical character came to be forgotten by Coulomb’s successors. [. . .] The way out of the difficulty lies in dropping the old fundamental principles and starting again from the elementary fact that sand consists of individual grains’. Ever since, Terzaghi’s words were not forgotten. The consideration of the behavior of geomaterials from the most basic units, i.e.,the particles, has in fact been the raison d’être of

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Proceedings of the 18th International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013

micromechanics of geomaterials. This discipline owes much of its success to the Discrete Element Method (DEM), introduced in the 1970s by Cundall & Strack (1970). DEM’s dominance in micromechanical analysis remains unsurpassed, with all but a few datasets at the particle scale arising from DEM simulations (e.g., Cundall 1989, Oda & Iwashita 2000, Thornton & Zhang 2006, Anthony 2007, Tordesillas 2007, Zhu et al. 2007, just to mention a few). As pertinently enunciated by Sibille & Froiio (2007): ‘This has led to the paradox of micromechanics of granular materials as a science based almost entirely on “virtual evidence”’.

On the experimental side, the past few years have witnessed some major advances, possibly heralding a new era in micromechanics of geomaterials. Measurements of contact forces, contacts and grain kinematics in 2D idealized assemblies of photoelastic discs have been achieved and analyzed (see the work of Behringer and co-workers, e.g., Kondic et al. 2012). Experimental 3D measurements on natural geomaterials are becoming possible at ever-increasing spatial resolution, and researchers are facing an unprecedented opportunity to integrate

r complement these measurements with data from DEM

simulations to probe the rheology of geomaterials, just as Terzaghi envisioned – from observations of behavior of individual particles.

o

Experimental access to information at the micro scale allows us to answer existing questions as well as to discover new mechanisms operating across the spatial scales, from the particle to the bulk. Of course, this new capability poses new challenges to modeling. Measurements at the small scale have an important role in revealing the physical origins of phenomena observed at the macro scale. However, rational theories are required to underpin this physics in terms of predictive tools, with numerical computations that extend the theoretical work, and allow for analysis of geomaterials with all their complexities, variabilities and uncertainties. While it is beyond any doubt that we can gain much from a more accurate description of these features at the finer scales, a fundamental issue (and the key challenge for the years to come) is to develop models capable of integrating information at multiple scales. This ambitious objective should be kept in mind as the background for the papers discussed in this report.

Table 1. List of papers belonging to this session.

keywords Authors Country Title

lab testing, sand, x-ray CT, compaction Otani J. et al. JapanFrance

Microscopic observation on compacted sandy soil using micro-focus X-ray CT

lab testing, sand, x-ray CT, strain localization Andò E. et al. France

Sweden Grain-scale experimental investigation of shear banding in sand

lab testing, clay, micrographs, creep, consolidation Yigit I. & Cinicioglu S.F. Turkey A look into time dependent behaviour

of clays at macro and micro scale lab testing, clay, chemical modification, soil improvement Minder P. & Puzrin A.M. Switzerland Microstructural changes leading to

chemically enhanced drainage

DEM, contact model, methane hydrates Jiang M.J. et al. China A Simplified Contact Model for Sandy Grains Cemented with Methane Hydrate

DEM, trapdoor, gravity flow, tunnel Kikkawa N. et al. JapanNew Zealand

Three dimensional discrete element simulation of trapdoor unloading and gravity flow of sandy granular material

DEM, small strain, shear wave velocity Ning Z. & Evans T.M. USADiscrete Element Method Study of Shear Wave Propagation in Granular Soil

DEM, computational fluid mechanics, dense phase flow Tomac I. & Gutierrez M. USA Particulate Modeling of Sand Slurry

Flow Retardation

analytical, effective stress equation Shao L.T. et al. China Uniform effective stress equation for soil mechanics

analytical, granular materials, crushing, abrasion, poly-disperse mixtures, compaction

Caicedo B. et al. Colombia USA

Modelling crushing of granular materials as a poly-disperse mixture

FEM, multi-scale modeling, wellbore damage Khoa H.D.V. et al. Norway

Macro- and micro-FE modelling of wellbore damage due to drilling and coring processes

lab testing, sand, plane strain, high pressure, methane hydrates Hyodo M. et al. Japan Shear strength and deformation of

methane hydrate bearing sand with fines

Lattice Boltzmann Method, relative permeability, petroleum geomechanics Pak A. & Sheikh B. Iran

Study of relative permeability variation during unsteady flow in saturated reservoir rock using Lattice Boltzmann method

lab testing, compacted soil, shear strength, constant water content, direct shear test

Heitor A. et al. Australia Behaviour of a compacted silty sand under constant water content shearing

lab testing, unsaturated soils, resilient modulus, thermo-hydro-mechanics Zhou C. & Ng C.W.W. Hong Kong

Experimental study of resilient modulus of unsaturated soil at different temperatures

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Technical Committee 105 / Comité technique 105

3 PAPERS PRESENTED TO THIS SESSION

3.1 Looking inside a soil sample using x-ray tomography

X-ray computed tomography (CT) is now widely used in material sciences and has amply proved its interest in geomechanics (see Viggiani & Hall 2012 for an overview). The principle of CT measurement consists of recording x-ray radiographs of a specimen at many different angular positions around the object. From these different projections, a three dimensional image of the object can be reconstructed with appropriate algorithms. X-ray CT is therefore a non-destructive imaging technique that allows quantification of internal features of a soil (or rock) sample in 3D.

Otani et al. (2013) used x-ray CT for imaging a sandy soil at different levels of compaction in 1D conditions. The motivation for this experimental study is to check whether the current criteria for quality control of dynamic compaction of soil for riverbanks are appropriate or not. Two cases were investigated, corresponding to a different number of blows to yield the same total compacting energy (cases 1 and 2 in Fig. 1, corresponding to higher and lower individual blow energy, respectively). Quantitative analysis of the 3D images from x-ray tomography allows Otani and coworkers to obtain the distribution of porosity in the sample and to follow its evolution with increasing compaction – see Fig. 1.

Figure 1. Spatial distribution of porosity in two samples of silty sand at different levels of compaction, as obtained with x-ray CT (Otani et al. 2013).

A second experimental study using x-ray tomography is presented by Andò et al. (2013). Further details and results can be found in Andò et al. (2012a, 2012b). The motivation for this study comes from the fact that strain localization presents major challenges for continuum models for geomaterials. For such models to be successful, the microstructure of the material (for sand, at the grain scale) should be explicitly taken into account, in one way or another, which in turn requires experimental characterization of shear banding at the grain scale. Andò et al. (2013) used x-ray tomography to image samples of two different sands (see Fig. 2) while they deform under triaxial compression.

Figure 2. Slices from x-ray images of angular Hostun sand (left) and rounded Caicos ooids (right) tested by Andò et al. (2013).

The results of this study clearly show that thanks to x-ray tomography, combined with either 3D Digital Image Correlation or Particle Tracking, the evolution of the 3D microstructure of a small sample of sand can be followed while it deforms, individual grains can be distinguished in the time-lapse 3D images, and analyzed to give the full 3D kinematics (displacement + rotation) of each individual grain in the sample (see as an example Fig. 3). Analysis of deformation at this scale is, in the Authors’ own words, a dream that has come through!

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Figure 3. Slices showing grains of a triaxial test on Hostun sand (top) and Caicos ooids (bottom) at 100 kPa confinement, colored by their vertical displacement and intensity of 3D rotation. The increments studied are highlighted on the stress-ratio vs. axial shortening in the middle of the figure (Andò et al. 2013).

3.2 Fine-grained soils from micro to macro

Yigit et al. (2013) present a contribution investigating the time dependent behavior of clays. In this experimental study, ESEM micrographs of kaolinite clay are taken under different levels of load in oedometric compression, and after different amounts of creep time. The pixel size in the micrographs was 8.4710-2 m,which is small enough to see the macro voids (see Fig. 4).

Figure 4. Raw (left) and segmented (right) ESEM micrographs of kaolinite clay (Yigit & Cinicioglu 2013).

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Quantitative analysis of the processed images reveals that the particle size distribution evolves in rather a surprising manner, and an attempt is made to correlate this to analytical creep parameters coming from Yin (1999). This is a nice example of analysis in which measurements at the micro scale are related to a macroscopic model. Further work might benefit from more robust image analysis techniques.

The study presented by Minder & Puzrin (2013) is extremely interesting and very well suited to this session. In order to achieve a macroscopic objective (increasing the permeability of a clay soil), the Authors develop an innovative experimental technique to modify the microstructure of the soil by using cation exchange. More precisely, the highly selective and strongly exchanging organic cation guanidinium was used to stabilize the interlayer distance between clay platelets. The effectiveness of this chemical treatment is characterized at the micro scale using SEM (see Fig. 5) and laser diffraction (see Fig. 6) as well as at the macro scale, where the improvement appears both in terms of increased permeability (see Fig. 7) and enhanced shear strength.

Figure 5. SEM-images of bentonite grains after washing in suspension with demineralised water. The calcium form remains finely dispersed (left), whereas the exposure to guanidinium ions (right) leads to the formation aggregates (Minder & Puzrin 2013).

Figure 6. Bimodal particle size distribution measured with laser diffraction. The volume fraction of the larger mode (aggregates) is significantly increased by the treatment (Minder & Puzrin 2013).

Figure 7. Decrease of hydraulic conductivity during sample compaction (including log-linear regression) of quartz-bentonite mixtures. For identical void ratio the modified soil is constantly about one order of magnitude more permeable (Minder & Puzrin 2013).

3.3 Learning mechanics from DEM

In the study by Jiang et al. (2013), DEM is used to describe at the particle level the mechanics of sand containing methane hydrates. In fact, the presence of methane hydrates in deep sea beds significantly alters the mechanical properties of the host sand material, because methane hydrates act as a bond between particles. The study by Jiang et al. (2013) introduces a simplified contact model (see Fig. 8), which was experimentally calibrated in the laboratory. The bond failure criterion is directly linked to the strength of methane hydrates, which depends on temperature, mean normal stress, density and methane hydrate saturation of the sand. The results of DEM simulations are compared to experimental results for the case of plane strain compression of a methane hydrate bearing sand; the results seem to show that although highly simplified, this model qualitatively captures the mechanical effects of cementation at the macro scale.

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Figure 8. Schematic illustration of (a) MH bonded soil grains and its response: (b) normal contact force Fn against overlap un; (c) shear contact force Fs against relative shear displacement us; and (d) contact moment M against relative rotation (Jiang et al. 2013).

Ground loss at the head of tunnels and in mining operations is a classical geotechnical problem with important implications for infrastructure development in urban settings. It has been studied at the laboratory scale for a long time; see for example the now classical trapdoor experiment by Terzaghi (1936). The paper by Kikkawa et al. (2013) reports a 3D DEM study of trapdoor unloading and gravity flow of granular material. The geometry of the problem studied is the same of the trapdoor experiments previously performed by Kikumoto & Kishida (2003) – see Fig.9a. Although the results of the DEM simulations agree well with the measurements from the actual experiments in some respects (for example, the vertical stress on the trapdoor when it is moved downward), DEM is substantially off target in other instances, for example in terms of the settlement of the surface of the sand above the trapdoor, see Fig. 9b. These “major discrepancies” are attributed by the Authors to the difference between the actual grains of Toyoura sand (used in the experiments) and the particles used in the DEM modeling (which are much larger and spherical). The general lesson to be learnt here is that the application of DEM to the analysis of boundary value problems is not trivial, especially when one seeks quantitative results and not only for a qualitative insight – calibration of the model remaining a major issue.

It has been often advocated that the use of DEM in micromechanical studies can significantly help advance our understanding of fundamental geomechanics (e.g., O’Sullivan 2011). In the writer’s opinion, DEM simulations are in fact a very useful tool for investigating the complex behavior of particulate materials, especially in conjunction with laboratory tests. In this respect, the study by Ning & Evans (2013) is of particular interest. The Authors address the fundamental issue of shear wave propagation in granular soil, using DEM simulations to investigate the effects of excitation frequency,

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particle size, and mean stress – which is of course possible, yet difficult, with laboratory experiments. Cylindrical assemblies of particles are subjected to shear wave excitation at one end and axial propagation velocities are measured (see Fig. 10). Micromechanical observations of the specimen are presented and analyzed in terms of particle velocity vectors, which highlight the complex motions of individual particles during wave propagation. As an example, velocity vectors in Fig. 11 show dominant S-wave motion (from right to left) in the central area of the specimen, while minor P-wave motion is observed on the sides (with the particle on the right moving downwards and the particle on the left moving upwards).

(a)

(b)

Figure 9. (a) Trap door and gravity flow testing apparatus (left), and DEM simulation (right); (b) Surface settlement of the sand above the center of the trapdoor when dt = 2.0 mm, layer thickness 150 mm (Kikkawa et al. 2013).

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Figure 10. DEM specimen with S-wave transmitting layer and receivers – for clarity, only half of the specimen is shown (Ning & Evans 2013).

Figure 11. Particle velocity vectors on different cutting planes of a DEM specimen at a 10ms time point after excitation (Ning & Evans 2013).

The paper by Tomac & Gutierrez (2013) also uses DEM as a tool for understanding processes, the focus of this nice study being the flow of dense sand slurries within a narrow channel – where “dense” means that the volumetric particle concentration is greater than 10%, and “narrow” that the width of the channel is less than 5 times the particle diameter. In these flow processes, clogging and velocity retardation often occur and are governed by sand concentration and slurry flow rate. The numerical model developed by the Authors couples the Discrete Element Method with computational fluid dynamics to study (in 2D) this flow process. A user-defined contact model is developed to capture the non-linear collision of submerged particles and walls. The theory of lubrication is also used to formulate a damping effect which is associated with the model. Some key results of this study are shown in Fig. 12. Maximum sand concentration (i.e., the concentration at which sand transport is not possible and the flow stops) is shown to depend on the ratio of channel width to particle diameter, as well as – to a lesser extent – on fluid pressure. Since solid and fluid phases have different average velocities, it is hard to average and come up with a unique slurry flow characterization at this point; the Authors conclude that a more comprehensive study is needed to address this issue.

Figure 12. (top) Clogging of sand in 4mm wide channel at initial volumetric sand concentration cv = 0.49 with particles velocities vectors in direction opposite to the flow; (middle) Unstable flow with formation of particles packs at initial cv = 0.39 in 4mm wide channel with fluid flow velocity vectors around packs; (bottom) Formation of particles packs at initial cv = 0.28 in 2mm wide channel (Tomac & Gutierrez 2013).

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3.4 Exploring micromechanics analytically

The contributions by Shao et al. (2013) and by Caicedo et al. (2013) both present theoretical developments. Shao et al. (2013)provide an analytical evaluation of the effect of pore water pressure on the effective stress in the soil skeleton for saturated and unsaturated media. Even though the developments are certainly correct, the objective and conclusions of this study appear somewhat obscure to the reporter.

The paper by Caicedo et al. (2013) reports an interesting theoretical development regarding grain crushing – which is of particular relevance when granular materials are used in engineering structures such as paved roads, railroads and highway embankments. The analytical model developed by the Authors aims to predict the evolution of the grain size distribution during loading. As an example, Fig. 13 shows the evolution of the grain size distribution predicted by the model after a very high number (up to one million) of loading cycles. The model uses the theory of poly-disperse mixtures proposed by De Larrard (2000), and predicts grain breakage by combining the geometrical relevance of a size class of grains with a statistical distribution of strength (given by a Weibull distribution). When a particle breaks, the size of its fragments is determined through a Markov process. The combination of these elements appears to capture grain breakage successfully – the application of this model to experimental results is shown to give very good agreement. It is the Authors’ contention that their model is a valid alternative to DEM, which can also deal with grain crushing but is computationally very expensive.

Figure 13. Evolution of the grain size distribution for different number of loading cycles (Caicedo et al. 2013).

3.5 Engineering applications for energy production

Interestingly, there is only one contribution to this session making use of the Finite Element Method (FEM). This is the numerical study by Khoa et al. (2013), which tackles the analysis of the damage induced by drilling and coring operations in the rock surrounding a wellbore. This is a two-scale analysis, in that a large scale 3D FE model is first used to simulate the stresses induced by drilling and coring (see Fig. 14); these stresses are then injected into a 2D micro scale FE model, the geometry of which is directly built on experimental SEM observations of well cemented sandstone. The results show that the micro FE model is able to pick up mechanisms of failure that simply do not feature in the macro-scale continuum model.

When going to the smaller scale analysis, the Authors implicitly assume that a Mohr-Coulomb elastic perfectly plastic model is capable of reasonably describing the stress state at the micro scale. It should be mentioned that the paper does not give any detail as for the determination of the mechanical parameters used in the analysis at the micro scale. Moreover, strong assumptions are made on the geological history (e.g., the grain skeleton carries the load before getting cemented – which

means that as soon as unloading occurs, the inter-granular cement gets loaded in tension and fails).

The approach adopted by Khoa and coworkers is certainly original and interesting. However, although the analysis presented in this study is indeed performed at two different scales, it should be stressed that the only link between the two scales is the stress in one point.

Figure 14. Full 3D FE modeling of different loads due to drill bit torque and axial load, mud-flow into formation and temperature change within one radius from wellbore wall (Khoa et al. 2013).

Figure 15. Cathodoluminescence SEM picture (top) from Storvoll & Bjorlykke (2004) and equivalent 2D micro FE model (bottom) used for studying induced damage around a wellbore during drilling and coring operations (Khoa et al. 2013).

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The second paper related to energy production is the study by Hyodo et al. (2013), who carried out a large laboratory experimental campaign to determine the mechanical properties and dissociation characteristics of sandy soils containing methane hydrate. The testing devices include a high pressure triaxial apparatus, capable of reproducing the in-situ conditions expected during methane extraction. Moreover, a high pressure and low temperature plane strain apparatus allowed the imaging of (localized) deformation of methane hydrate bearing sand due to methane hydrate production. Using this latter apparatus, two tests were performed: plane strain compression and methane hydrate dissociation by depressurization. The dissociation tests were carried out in two different ways, either prescribing a pore water pressure history corresponding to real production of methane hydrate (Case 1), or by simulating the stress conditions in the vicinity of the production well, where the material is close to failure (Case 2).

This is a very interesting application, in which the micro-scale physics (the methane hydrate bonding) is clearly driving the behavior observed at the macro scale. The study certainly lacks direct experimental observations at the micro scale – even though the strain fields obtained by DIC can be considered as measurements at a scale somewhere in between macro and micro. By controlling the pressure (see Fig. 16) Hyodo and co-workers nicely isolate the effect at the macro scale of a micro-scale feature (cementation); in this respect, their contribution fits very well the general theme of the session: from micro to macro.

Fig.16. Effective stress ratio vs. axial strain during a dissociation test for Case 2. Point (a) corresponds to the point before dissociation, and point (b) to the point when pwp is decreased from 10MPa to 3MPa. Point (c) corresponds to the point when MH is dissociated, and point (d) to the point when the specimen failed due to an increase in pore water pressure (re-pressurization) – note that failure occurred when the stress path reached the strength of the host sand (Hyodo et al. 2013).

Pak & Sheikh (2013) present the 2D implementation of a Lattice Boltzmann Method code that allows the simulation of immiscible fluids (water and oil) flowing in porous rock – a classical and central issue in petroleum geomechanics. The code is validated against some basic test cases, and is then applied to partially reproduce experimental results from Valavanides et al.(1998) and Tsakiroglou et al. (2007) in both steady and unsteady conditions. The comparison between experimental and simulated results is encouraging and shows that the technique is promising (see Figs. 17 and 18).

3.6 Contributions focusing on the macro scale

Heitor et al. (2013) present an experimental study on unsaturated silty sands, where materials compacted to different levels are tested at different constant water contents in direct shear. Even though the results of this experimental investigation

are justified by means of micromechanical arguments, this remains a study at the macro scale only.

The contribution by Zhou et al (2013) deals with an experimental evaluation of the effects of suction and temperature on the cyclic behavior of soils – in particular the resilient modulus. This is also a study at the macro scale only.

Figure 17. (left) example of initial distribution of the fluids in steady-state simulation; (right) example of invasion of wetting fluid(green) in unsteady state simulation (Pak & Sheikh 2013).

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Figure 18. Comparison of Lattice Boltzmann modeling results and experimental relative permeability curves (steady state) (Pak & Sheikh 2013).

4 CONCLUSIONS

While our discipline has traditionally focused on the macro scale, the papers submitted to this session show that a broad spectrum of approaches, materials and techniques are now becoming available to explore the behavior of geomaterials at smaller scales. These tools are bringing new insights into the physical processes driving the macroscopic behavior.

As our understanding of the micro behavior of geomaterials progresses, it will be crucial to explicitly build a link between the micro and macro scales – none of the contributions to this session has yet accomplished this ambitious goal. As a matter of fact, bridging the gap between micromechanical studies and continuum approaches at the macro scale is one of the emergent directions in mechanics and material science, not only in geomechanics. This should be the central theme for discussion in this session.

Multiscale methods have emerged recently in geomechanics to bridge different material scales ranging from the micro scale to continuum scale (e.g., Andrade et al. 2011, Nitka et al. 2011, Frey et al. 2013). These methods aim at obtaining constitutive responses at the continuum scale, without resorting to phenomenology. However, in most of these studies the multi-scale character is essentially tackled at the modeling level, involving theoretical and computational developments but missing experimental input. In the writer’s opinion, experiments should be an essential part of a multiscale approach – how else can one make a realistic connection between micro scale physics and macro scale responses?

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5 ACKNOWLEDGEMENTS

I am deeply indebted to my young and smart colleagues Edward Andò and Alessandro Tengattini in Grenoble for their generous and invaluable help with this report.

6 REFERENCES

6.1 Papers belonging to this sessions

Andò E., Desrues J., Bésuelle P., Viggiani G., Hall S.A. 2013. Un rêve devenu réalité : explorer une bande de cisaillement à l’échelle des grains. Proc. of the 18th ICSMGE, Paris.

Caicedo B., Ocampo M., Vallejo L. 2013. Modelling crushing of granular materials as a poly-disperse mixture. Proc. of the 18th

ICSMGE, Paris.Heitor A., Rujikiatkamjorn C., Indraratna B. 2013. Behaviour of a

compacted silty sand under constant water content shearing. Proc. of the 18th ICSMGE, Paris.

Hyodo M., Yoshimoto N., Kato A., Yoneda J. 2013. Shear strength and deformation of methane hydrate bearing sand with fines. Proc. of the 18th ICSMGE, Paris.

Jiang M.J., Liu F., Zhu F., Xiao Y. 2013. A Simplified Contact Model for Sandy Grains Cemented with Methane Hydrate. Proc. of the 18th ICSMGE, Paris.

Khoa H.D.V., L. Grande L., Jostad H.P. 2013. Macro- and micro-FE modelling of wellbore damage due to drilling and coring Processes. Proc. of the 18th ICSMGE, Paris.

Kikkawa N., Itoh K., Toyosawa Y., Pender M.J., Orense R.P. 2013. Three dimensional discrete element simulation of trapdoor unloading and gravity flow of sandy granular material. Proc. of the 18th ICSMGE, Paris.

Minder P. & Puzrin A.M. 2013. Microstructural changes leading to chemically enhanced drainage. Proc. of the 18th ICSMGE, Paris.

Ning Z. & Evans T.M. 2013. Discrete Element Method Study of Shear Wave Propagation in Granular Soil. Proc. of the 18th ICSMGE, Paris.

Otani J., Mukunoki T., Takano D., Chevalier B. 2013. Microscopic observation on compacted sandy soil using micro-focus X-ray CT. Proc. of the 18th ICSMGE, Paris.

Pak A. & Sheikh B. 2013. Study of relative permeability variation during unsteady flow in saturated reservoir rock using Lattice Boltzmann method. Proc. of the 18th ICSMGE, Paris.

Shao L.T., Liu G., Guo X.X. 2013. Uniform effective stress equation for soil mechanics. Proc. of the 18th ICSMGE, Paris.

Tomac I. & Gutierrez M. 2013. Particulate Modeling of Sand Slurry Flow Retardation. Proc. of the 18th ICSMGE, Paris.

Zhou C. & Ng C.W.W. 2013. Experimental study of resilient modulus of unsaturated soil at different temperatures. Proc. of the 18th

ICSMGE, Paris. Yigit I. & Cinicioglu S.F. 2013. A look into time dependent behaviour

of clays at macro and micro scale. Proc. of the 18th ICSMGE, Paris.

6.2 Other papers cited in this report

Andò E. Hall S.A., Viggiani G., Desrues J., Bésuelle P. 2012a. Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotechnica, 7, 1, 1-13.

Andò E., Hall S.A., Viggiani G., Desrues J., Bésuelle P. 2012b. Experimental micromechanics: grain-scale observation of sand deformation. Géotechnique Letters, Vol. 2, Issue July-September, 107 –112.

Andrade J.E., Avila C.F., Hall S.A., Lenoir N., Viggiani G. 2011. Multiscale modeling and characterization of granular matter: From grain kinematics to continuum mechanics. Journal of the Mechanics and Physics of Solids 59, 237–250.

Antony S.J. 2007. Link between single-particle properties and macroscopic properties in particulate assemblies: role of structures within structures. Phil. Trans. R. Soc. A 365, 2879–2891.

Cundall P. 1989. Numerical experiments on localization in frictional materials. Ingenieur-Archiv 59, 148–159.

Cundall P.A & Strack O.D.L. 1979. A discrete numerical model for granular assemblies. Géotechnique 29, 47–65.

Dascalu C., Cambou B. (Editors) 2008. Multiscale approaches to geomaterials. Acta Geotechnica 3 (3).

De Larrard F. 2000. Compacité et homogénéité des mélanges granulaires. In: Structures Granulaires et Formulation des Bétons, LCPC.

Frey J., Chambon R., Dascalu C. 2013. A two-scale poromechanical model for cohesive rocks. Acta Geotechnica 8 (2) 107-124.

Kikumoto M. & Kishida K. 2003. Mechanical behavior on the sandy ground through the 3-D trapdoor experiment, Proc. of the 12th

Asian Regional Conference on Soil Mechanics & Geotechnical Engineering.

Kondic L., Goullet A., O'Hern C.S., Kramar M., Mischaikow K., Behringer R.P. 2012. Topology of force networks in compressed granular media, Europhysics Letters 97, 54001.

Oda M. & Iwashita K. 2000. Study on couple stress and shear band development in granular media based on numerical simulation analyses. Int. J. Eng. Sci. 38, 1713–1740.

O’Sullivan C. 2011. Particle-based Discrete Element Modelling: Geomechanics Perspective. International Journal of Geomechanics, ASCE, 11 (6), 449-464.

Nitka M., Combe G., Dascalu C., Desrues J. 2011. Two-scale modeling of granular materials: a DEM-FEM approach. Granular Matter 13 (3), 277-281.

Santamarina J.C. 2003. Soil behavior at the microscale: particle forces. Proc. Symp. Soil Behavior and Soft Ground Construction – The Ladd Symposium, October 2001, MIT, Boston, ASCE Special Publications 119, 25-56.

Sibille L. & Froiio F. 2007. A numerical photogrammetry technique for measuring microscale kinematics and fabric in Schneebeli materials. Granular Matter 9, 183–193.

Storvoll V. & Bjorlykke K. 2004. Sonic velocity and grain contact properties in reservoir sandstones. Petroleum Geoscience 10 (3), 215-226.

Terzaghi K. 1920. Old earth pressure theories and new test results. Eng. News Record 85, 632–637.

Terzaghi K. 1936. Stress distribution in dry and saturated sand above a yielding trap-door. Proceedings of the1st International Conference of Soil Mechanics, Harvard University, Cambridge (USA), 1, 307-311.

Thornton C. & Zhang L. 2006. A numerical examination of shear banding and simple shear noncoaxial flow rules. Philos. Mag. 86, 3425–3452.

Tordesillas A. 2007. Force chain buckling, unjamming transitions and shear banding in dense granular assemblies. Philos. Mag. 87, 4987–5016.

Tsakiroglou C.D., Avraam D.G., Payatakees A.C. 2007. Transient and steady-state relative permeabilities from two-phase flow experiments in planar pore networks. Advances in Water Resources,30, 1981–1992.

Valavanides M.S., Constantinides G.N., Payatakes A.C. 1998. Mechanistic Model of Steady-State Two-Phase Flow in Porous Media Based on Ganglion Dynamics. Transport in Porous Media,30, 267-299.

Viggiani G., Hall S.A. 2012. Full-field measurements in experimental geomechanics: historical perspective, current trends and recent results. In: Advanced experimental techniques in geomechanics, G. Viggiani, S.A. Hall and E. Romero Eds., 3-67.

Yin J.H. 1999. Non-linear creep of soils in oedometer tests. Géotechnique 49 (5), 699-707.

Zhu H.P., Zhou Z.Y., Yang R.Y., Yu A.B. 2007. Discrete particle simulation of particulate systems: theoretical developments. Chem.Eng. Sci. 62, 3378–3396.

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Un rêve devenu réalité : explorer une bande de cisaillement à l’échelle des grains

Grain-scale experimental investigation of shear banding in sand

Andò E., Desrues J., Bésuelle P., Viggiani G. UJF/Grenoble INP/CNRS, Laboratoire 3SR, Grenoble, France

Hall S. Division of Solid Mechanics, Lund University, Lund, Sweden and European Spallation Source AB, Lund, Sweden

RÉSUMÉ : La caractérisation expérimentale du comportement des milieux granulaires repose depuis la fondation de la mécanique des sols sur des expériences de laboratoire réalisées sur des échantillons suffisamment grands vis-à-vis des grains, pour être considérés comme représentatifs du comportement moyen du matériau. Le développement des modèles aux éléments discrets (DEM) a permis de simuler numériquement le comportement d’assemblées de particules. Mais à l’exception de quelques études pionnières, il manquait jusqu’à récemment à la micro-mécanique des milieux granulaires numérique son pendant expérimental. Grâce aux progrès récents de la micro-tomographie à rayons X, il est aujourd’hui possible, avec un synchrotron comme l’ESRF mais aussi avec un tomographe delaboratoire tel que celui installé au laboratoire 3SR, de réaliser, sous tomographie, des expériences de mécanique des sols tels que des essais triaxiaux sur sable, avec une résolution spatiale permettant de suivre très précisément le mouvement de chacun des grains de l’échantillon. L’apparition de bandes de cisaillement et les mécanismes intimes qui y conduisent peuvent ainsi être explorés expérimentalement, et confrontés aux résultats de la modélisation numérique.

ABSTRACT: Strain localization presents major challenges for continuum models for geomaterials. For such models to be successful, the microstructure of the material (for sand, the grain structure) should be explicitly taken into account, in one way or another, whichin turn requires experimental characterization of shear banding at the grain scale. In this paper, x-ray tomography is used to image samples of two different sands while they deform under triaxial compression. The kinematics (displacement + rotation) of eachindividual grain in the sample is measured with different techniques, combining recent developments in image correlation and particle tracking. A few selected results are presented, which allow a number of interesting observations to be made on shear bands when theyare fully developed as well as at the transition from homogeneous to localized deformation.

KEYWORDS: Micro-Mechanics, Sand, Grains, Micro Tomography, CT-Scan, Image Analysis, Strain localization, triaxial test

1 INTRODUCTION

La localisation de la déformation, qui est associée généralement à la rupture, se joue à une échelle proche de celle des grains. La modélisation tend actuellement à introduire des éléments de micro-structure pour mieux modéliser la localisation. Il en découle un besoin de données expérimentales à cette échelle.

Pour les milieux granulaires, la micro-échelle naturelle est celle des grains. Des études ont été faites sur les mécanismes de déclenchement de la localisation à cette échelle (e.g., Mühlhaus and Vardoulakis1987, Iwashita and Oda 2000). Historiquement, le comportement du sable à cette échelle a été exploré plutôt numériquement (par exemple par la méthode des éléments discrets) qu’expérimentalement.

Les techniques expérimentales en géomécanique ont évolué considérablement ces dernières années (voir une revue de synthèse par Viggiani et Hall (2008) pour plus de références). La tomographie à rayons X permet aujourd’hui de suivre expérimentalement l’évolution de la microstucture complète en 3D lors d’un essai triaxial sur un petit échantillon de sable (échantillon de 60.000 grains qui reste néanmoins représentatif). On distingue parfaitement chacun des grains, avec une résolution telle que l’on peut les suivre individuellement dans leur mouvement au cœur de l’échantillon. Pour cela on utilise la Corrélation d’Image Numérique (Hall et al. 2010) ou le suivi de particules (Andò et al. 2012a), ou plus récemment, une combinaison des deux (Andò et al. 2012b).

On présente ici une sélection de résultats obtenus sur deux sables différents, l’un anguleux (sable d’Hostun) et l’autre

arrondi (Caicos ooids), en utilisant le tomographe à rayons X du laboratoire 3SR de Grenoble. On s’intéresse à la naissance et à l’évolution d’une bande de cisaillement dans un échantillon soumis à un essai triaxial sous tomographie. L’analyse est faite sur l’ensemble de l’échantillon, et aussi sur un cube contenant un millier de grains, pris à proximité de la bande de cisaillement. Dans les deux cas on s’intéresse à la cinématique fine de cet assemblage de grains, chacun étant caractérisé par son déplacement et sa rotation en 3D.

2 MÉTHODES

2.1 Dispositifs expérimentaux et matériaux d’essai

Les essais décrits ici sont des essais triaxiaux classiques en compression, à ceci près que les échantillons sont très petits : diamètre 11 mm, hauteur 24 mm. Le système triaxial a été mis en œuvre précédemment sur une ligne de lumière synchrotron de l’ESRF, il est décrit en détail dans la référence Hall et al. (2010). Tout comme l’échantillon, le système est de type classique et se distingue surtout par sa taille réduite ; on notera cependant que la cellule est réalisée en PMMA, et que pour éviter d’interférer avec le faisceau lors de la rotation du montage devant celui-ci (un tour complet pour réaliser un millier de radiographies), on a supprimé les tirants habituels ; de ce fait c’est le corps de la cellule qui reprend en traction l’effort de compression sur l’échantillon. L’effort est appliqué par un vérin mécanique contrôlé par un moteur.

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La figure 1 présente à gauche la source à rayons X, l’échantillon dans la cellule fixée sur le plateau tournant, et à droite l’imageur qui permet d’enregistrer les radiographies du montage pour chaque nouvelle incidence (e.g. vignette en bas à droite). Le dispositif de chargement est placé sous le plateau tournant, il applique un déplacement vers le haut de la tête inférieure alors que la tête supérieure est fixe.

Figure 1. Installation d’essai triaxial sur sable “in situ” c’est à dire sous faisceau dans le tomographe à Rayons X du laboratoire 3SR.

L’étude présentée porte sur deux sables (voir Fig. 2), l’un est le sable d’Hostun HN31, anguleux, quartzique, avec un D50 de 338 µm, et l’autre le Caicos ooids (sable d’ooïdes), un sable naturel provenant de British West Indies, constitué de grains de calcite arrondis, avec un D50 de 420 µm. Les échantillons sont déposés par pluviation dans un moule garni d’une membrane en latex fine, ce qui produit un état initial dense. Les échantillons sont testés en état sec, sous pression de cellule constante, avec une vitesse de raccourcissement axial de 21 µm/min. L’essai est interrompu à différentes étapes, pour permettre la réalisation d’une tomographie qui prend environ deux heure. La résolution des images 3D volumiques obtenues est de 15,6 µm/voxel (pixel en 3D, cubique).

2.2 Mesure de la cinématique à l’échelle des grains

Pour étudier la déformation à l’échelle des grains, il faut réussir à identifier chaque grain dans les images 3D qui résultent de la tomographie. Il faut donc binariser les images, c’est-à-dire décider pour chaque voxel, en comparant son niveau de gris à un certain seuil, s’il fait partie des grains ou de l’espace poral. Il faut ensuite séparer l’espace identifié comme celui des grains, en grains individuels : c’est la segmentation, qui utilise un algorithme dit de « partage des eaux » (watershed). Une fois séparés, on identifie chaque grain dans l’image i par un label qui lui est propre. Enfin on procède au suivi de grain entre deux images 3D, en recherchant l’homologue de chaque grain de l’image i dans l’image i+1 dans un certain voisinage spatial. L’élu est le grain dont la « carte d’identité géométrique » est la plus proche. La caractéristique géométrique considérée est le volume. Ce processus est très rapide, car il repose non pas sur une corrélation mais sur une simple recherche dans une liste de grains. La méthode a été baptisée « ID-Track ». Le déplacement est défini comme le changement de position du centre de masse du grain d’une configuration à l’autre. L’erreur de mesure est

estimée à moins de 1.5 µm dans les conditions considérées ici (voir Andò et al. 2012a).

En ce qui concerne la rotation 3D des grains, dans une première approche nous l’avons mesurée en déterminant le changement d’orientation de directions caractéristiques du grain, en substance les valeurs propres majeure et mineure du tenseur de moment d’inertie du volume occupé par le grain. Cette approche permet des calculs très rapides, car elle repose sur la comparaison assez simple des orientations de deux vecteurs 3D. En revanche, l’expérience a montré que cette approche conduit à un taux excessif de mesures aberrante

Figure 3. Résultats de ID-Track pour cinq incréments de compression triaxiale sur sable d’Hostun et sur Caicos Ooids. Ligne supérieure : déplacement ; ligne inférieure : rotation. Les incréments 1-2, 2-3,… réfèrent aux états indiqués 1, 2, 3,… sur les courbes contrainte-déformation présentées en vignette.

s, quFigure 2. Deux coupes 2D dans les volumes 3D tomographiques des

échantillons de sable Hostun HN31 et Caicos Ooids ’on attribue soit à une définition trop fragile des vecteurs

propres du tenseur (lorsque celui-ci est trop proche de l’isotropie, c’est-à-dire le grain trop sphérique, ou de l’isotropie transverse, grain à section circulaire), soit à des sauts de 180 degrés, qu’on peut certes traiter mais non sans complications.

Une nouvelle approche hybride a été définie (Andò et al. 2012b), sur une base de corrélation d’image numérique à l’échelle du grain, associée à une poursuite de grain suivant la méthode précédemment mentionnée. La méthode de corrélation à l’échelle du grain, dans le même esprit que Hall et al. 2010, exploite au mieux la richesse de l’information concernant le grain (il contient typiquement plusieurs milliers de voxels) plutôt que de la résumer à trois vecteurs d’orientation. En revanche, elle est plus coûteuse en temps de calcul. L’approche hybride coupe court à la recherche coûteuse des grains homologues par corrélation d’image, en utilisant la méthode ID-Track pour établir les correspondances entre grains. Pour la méthode de corrélation d’image particulaire, un grain définit dans chacune des deux images une zone d’intérêt. On va appliquer une transformation, de type translation et rotation, à la zone d’intérêt de l’image de référence, pour que cette zone d’intérêt vienne s’identifier au mieux à son homologue dans la seconde image. La transformation géométrique est basée sur une interpolation trilinéaire, et la recherche de la meilleure

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ressemblance est menée selon un algorithme d’optimisation du type BFGS. On obtient ainsi 3 composantes de déplacement et trois de rotation pour le grain considéré, cette procédure est

échantillon. En termes de temps end environ 10

sec

soit en fonction de son déplacement (m

lacement. En

dans un arrangement fortement imbriqué comme le sont naturellement les grains de

répétée pour tous les grains de l’de calcul, sur un Intel Core i7, la corrélation pr

ondes par grain, le temps de produire en moyenne 200 images déplacées et tournées avant de trouver un minimum.

3 QUELQUES RÉSULTATS

La Figure 3 montre les résultats de l’application de la méthode de poursuite de grains ID-Track aux cinq premiers incréments de deux essais triaxiaux menés sur les deux sables précités. Les images présentent des coupes axiales dans le volume de grains dans l’état de référence. L’orientation des coupes a été choisie telle qu’elles contiennent à la fois l’axe de l’échantillon, et la normale à la bande qui se forme lors de l’essai. Pour chaque incrément, chaque grain suivi a été coloré, dans cette illustration,

odule du vecteur déplacement), soit de la valeur de sa rotation (l’axe de rotation n’est pas représenté). Les grains qu’on n’a pas réussi à suivre sont laissés en blanc. On rappelle que le déplacement est nul en tête d’échantillon, en raison du chargement par le bas.

Le comportement global est clair : le premier incrément est diffus, une légère déviation de la verticalité de l’échantillon vierge produit une légère inclinaison du champ de dép

mécanique des milieux continus, ceci se traduirait par un champ de déformation pas tout à fait uniforme, mais sans gradients forts. Dès l’incrément 2-3, le champ montre un gradient plus marqué dans la direction de la future bande de cisaillement, qui devient évidente à l’incrément 4-5.

En ce qui concerne les rotations, le pic (état 3) marque une transition entre l’absence d’organisation spatiale claire, et une concentration dans la zone de déformation localisée. On note toutefois que la zone de concentration de rotation paraît plus large que la bande de cisaillement vue sur les cartes de déplacement ; on relève une largeur de 10 à 12 grains dans la zone de rotations fortes dans le sable d’Hostun, moins (7 à 8) dans Caicos ooids (voir Figure 4). La bande est aussi plus inclinée dans ce dernier cas. On peut penser que la différence de forme de grains est responsable de cette différence de largeur de bande de forte rotation, en raison de la plus grande distance de transmission de la rotation d’un grain

Figure 5. Haut : courbes contrainte-déformation, sable Caiocos Ooids. Bas : coupes axiales des volumes tomographiques au début et à la fin du test, montrant l'emplacement des grains analysés figure 6.

forme irrégulière et allongée, que dans un arrangement de grainsronds qui peuvent tourner individuellement en glissant par rapport aux voisins. Ceci peut aussi expliquer la plus granderésistance résiduelle du sable anguleux.

Figure 6. Haut et milieu: déplacement vertical et rotation des grains du petit volume 3D extrait comme indiqué en figure 5. Les valeurs négatives de déplacement signifient 'vers le haut'. ligne du bas : rotations des grains sur une coupe global de l'échantillon Figure 4. Zoom sur la bande de cisaillement, incrément 5-6

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On présente Figure 5 la courbe contrainte-déformation pour un autre essai sur Caicos ooids. La variation de volume est essentiellement et nettement dilatante, après une brève phase de légère contraction. Les phases de relaxation observables sur la courbe de contrainte correspondent aux étapes de scan, durant lesquelles l’écrasement axial est stoppé pendant environ une heure ; elles sont au nombre de 16. Les incréments analysés ci-après sont signalés sur la courbe : un incrément en tout début d’essai, l’incrément juste avant le pic de résistance, et un incrément en palier, alors que la variation de volume globale a complètement cessé. En bas de la figure 5, on montre un sous-volume de grains sélectionnés dans la configuration initiale, et ce que deviennent ces grains en phase finale.

La Figure 6 (haut et milieu) montre l’évolution du sous-volume au cours des trois incréments choisis. Un millier de grains constituent ce volume, dont on représente la position en début d’incrément avec le déplacement vertical au cours de l’incrément codé en couleur (ligne du haut), et la rotation de même (ligne du milieu). Les déplacements sont négatifs parce qu

gradient de déplacement très concentré. On observe la coexistence d’une

us-volume), et tations autour de la

s échantillons de matériaux à des rés

elle capacité d’imagerie, avec un sav

es mécanismes aussi bien lors de la phase initiale de montée rapide

de

et esprit ont été effectués en 2D par Calvetti et al. (1997) ; le passage au

cisaillement repose sur une approche de bifurcation, qui fait

ment du matériau un rôle clé dans la tinuité spatiale de la vitesse de

D et leurs caractéristiques, par exemple l’épaisseur reliée à la taille des

tants. En effet la localisation de la souvent les ouvrages géotechniques, tels

bar

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echanics of Cohesive-Frictional Materials, 2, 121–163

Hall, S.A., Bornert, M., Desrues, J., Pannier, Y., Lenoir, N., Viggiani, G., and Bésuelle, P. (2010) Discrete and continuum experimental study of localised deformation in Hostun sand under triaxial compression using x-ray μCT and 3D digital image correlation. Géotechnique, 60(5), 315–322.

Iwashita, K., and Oda, M. (2000) Micro-deformation mechanism of shear banding process based on modified distinct element method. Powder Technology, 109, 192–205.

Mühlhaus, H.B., and Vardoulakis, I. (1987) The thickness of shear bands in granular materials. Géotechnique, 37(3), 271-283.

Sibille, L., and Froiio, F. (2007) A numerical photogrammetry technique for measuring microscale kinematics and fabric in Schneebeli materials. Granular Matter, 9, 183–193.

Viggiani G., and Hall, S.A. (2008) Full-field measurements, a new tool for laboratory experimental geomechanics. Keynote paper in: Proceedings of the 4th International Symposium on Deformation Characteristics of Geomaterials, IOS Press, 1, 3-26.

e dirigés vers le haut, puisque l’écrasement de l’échantillon est appliqué par le bas. Un déplacement de 1 pixel vaut 15,6 µm. Dans le premier incrément, les déplacements correspondent à une déformation diffuse, et les rotations sont insignifiantes et non organisées spatialement. Dans l’incrément 2, les déplacements restent diffus mais leur gradient n’est plus vertical, ce qui indique une rotation globale ; c’est le résultat d’une bande de cisaillement naissante, régnant sur une large zone, qu’on observe macroscopiquement. Les rotations individuelles sont beaucoup plus grandes, mais toujours pas organisées spatialement. L’incrément 3 montre une bande de cisaillement complètement développée, nettement plus étroite que la dimension du sous-volume comme le révèle le

bande et de blocs quasi rigides au sein du soconfirme la concentration des grandes robande, plus fortes en son milieu mais la débordant assez nettement comme déjà observé plus haut.

La ligne du bas de la Figure 6 montre l’intensité des rotations dans des coupes verticales de l’échantillon complet, aux différents incréments. Les incréments 1 et 3 confirment les observations faites sur les sous-volumes, à savoir absence d’organisation dans le premier incrément, et forte concentration dans la bande dans l’incrément final. Cependant l’incrément 2, situé au pic, montre que les rotations sont organisées à l’échelle de l’échantillon, dans une bande large d’une quinzaine de grains, centrée sur ce qui deviendra la bande de cisaillement développée, nettement plus étroite, dans les incréments suivants. Il faut noter que l’observation des sous-volumes seulement ne permet pas de détecter cette phase d’organisation globale précoce, car elle se déroule à l’échelle de l’échantillon.

4 CONCLUSION ET PERSPECTIVES

Les développements récents de la micro-tomographie à rayons X ont rendu possible d’imager en 3D de manière non destructive et non invasive de

olutions qui étaient tout simplement inimaginables voici vingt ans. Dans le contexte de la mécanique des sols, c’est un vieux rêve qui est devenu réalité : suivre individuellement tous les grains d’un échantillon en cours de déformation. La combinaison de cette nouv

oir-faire expérimental permettant de mener à bien des essais mécaniques représentatifs sous faisceau RX d’une part, et un savoir-faire numérique en analyse par corrélation d’image d’autre part, est extrêmement puissante. Nous sommes capables aujourd’hui d’extraire du volume énorme de données produites par ces essais, de nouveaux éléments d’analyse des mécanismes de déformation à l’œuvre au cœur des échantillons, tout au long d’un essai. On peut ainsi observer l’évolution de c

la résistance mobilisée, puis pendant que la déformation diffuse se développe, puis se localise et conduit finalement à un plateau caractérisé par une résistance résiduelle dégradée par rapport à la résistance de pic.

Un projet long terme qui découle naturellement de ces observations consiste à étudier l’évolution des contacts et leur rôle dans les mécanismes observés. Des travaux dans c

3D est désormais possible. D’un point de vue théorique, l’étude des bandes de

jouer à la loi de comporterecherche d’une discondéformation. Cependant, c’est seulement avec des approches en milieux enrichis que la notion d’épaisseur de bande de cisaillement émerge des équations. Les données expérimentales produites ici sont pertinentes pour établir de telles approches.

Du point de vue de l’application, les résultats concernant le développement des bandes de cisaillement en 3

grains, sont impordéformation affecte qu’excavations et pentes naturelles ou artificielles, digues et

rages, tunnels et galeries, forages, sites de stockage.

arque : une version en couleur de ce texte est disponible : ://l3sphnum.hmg.inpg.fr/HomepageS1/etagere/ICSMGE_Ando.pdf

REMERCIEMENTS

Ces études ont été rendues possibles par le financement attribué par l’ANR « blanche » à deux projets successifs, MicroMODEX

05-2008) et GeoBridge (2009-2013) qui ont permis de créer stallation de tomographie et de rassembler ou développer

autour d’elle les compétences et outils nécessaires.

RÉFÉRENCES

Alonso-Marroquin, F., and Vardoulakis, I. (2005) Micromechanics of shear bands in granular media. Powders and Grains, 701–704. ò, E., Hall, S.A., Viggiani, G., Desrues, J., and Bésuelle, P. (2012a) Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotechnica, 7(1), 1-13.

Andò, E., Hall, S.A., Viggiani, G., Desrues, J., and Bésuelle, P. (2012b) Experimental micromechanics: grain-scale observation of sand deformation. Géotechnique Letters, http://www.icevirtuallibrary.com/content/serial/geolett.

vetti, F., Combe, G., and Lanier, J. (1997) Experimental micromechanical analysis of a 2D granular material: relation between structure evolution and loading path. M

1003

On présente Figure 5 la courbe contrainte-déformation pour un autre essai sur Caicos ooids. La variation de volume est essentiellement et nettement dilatante, après une brève phase de légère contraction. Les phases de relaxation observables sur la courbe de contrainte correspondent aux étapes de scan, durant lesquelles l’écrasement axial est stoppé pendant environ une heure ; elles sont au nombre de 16. Les incréments analysés ci-après sont signalés sur la courbe : un incrément en tout début d’essai, l’incrément juste avant le pic de résistance, et un incrément en palier, alors que la variation de volume globale a complètement cessé. En bas de la figure 5, on montre un sous-volume de grains sélectionnés dans la configuration initiale, et ce que deviennent ces grains en phase finale.

La Figure 6 (haut et milieu) montre l’évolution du sous-volume au cours des trois incréments choisis. Un millier de grains constituent ce volume, dont on représente la position en début d’incrément avec le déplacement vertical au cours de l’incrément codé en couleur (ligne du haut), et la rotation de même (ligne du milieu). Les déplacements sont négatifs parce qu

gradient de déplacement très concentré. On observe la coexistence d’une

us-volume), et tations autour de la

s échantillons de matériaux à des rés

elle capacité d’imagerie, avec un sav

es mécanismes aussi bien lors de la phase initiale de montée rapide

de

et esprit ont été effectués en 2D par Calvetti et al. (1997) ; le passage au

cisaillement repose sur une approche de bifurcation, qui fait

ment du matériau un rôle clé dans la tinuité spatiale de la vitesse de

D et leurs caractéristiques, par exemple l’épaisseur reliée à la taille des

tants. En effet la localisation de la souvent les ouvrages géotechniques, tels

bar

Rem http

5

(20l’in

6

And

Cal

echanics of Cohesive-Frictional Materials, 2, 121–163

Hall, S.A., Bornert, M., Desrues, J., Pannier, Y., Lenoir, N., Viggiani, G., and Bésuelle, P. (2010) Discrete and continuum experimental study of localised deformation in Hostun sand under triaxial compression using x-ray μCT and 3D digital image correlation. Géotechnique, 60(5), 315–322.

Iwashita, K., and Oda, M. (2000) Micro-deformation mechanism of shear banding process based on modified distinct element method. Powder Technology, 109, 192–205.

Mühlhaus, H.B., and Vardoulakis, I. (1987) The thickness of shear bands in granular materials. Géotechnique, 37(3), 271-283.

Sibille, L., and Froiio, F. (2007) A numerical photogrammetry technique for measuring microscale kinematics and fabric in Schneebeli materials. Granular Matter, 9, 183–193.

Viggiani G., and Hall, S.A. (2008) Full-field measurements, a new tool for laboratory experimental geomechanics. Keynote paper in: Proceedings of the 4th International Symposium on Deformation Characteristics of Geomaterials, IOS Press, 1, 3-26.

e dirigés vers le haut, puisque l’écrasement de l’échantillon est appliqué par le bas. Un déplacement de 1 pixel vaut 15,6 µm. Dans le premier incrément, les déplacements correspondent à une déformation diffuse, et les rotations sont insignifiantes et non organisées spatialement. Dans l’incrément 2, les déplacements restent diffus mais leur gradient n’est plus vertical, ce qui indique une rotation globale ; c’est le résultat d’une bande de cisaillement naissante, régnant sur une large zone, qu’on observe macroscopiquement. Les rotations individuelles sont beaucoup plus grandes, mais toujours pas organisées spatialement. L’incrément 3 montre une bande de cisaillement complètement développée, nettement plus étroite que la dimension du sous-volume comme le révèle le

bande et de blocs quasi rigides au sein du soconfirme la concentration des grandes robande, plus fortes en son milieu mais la débordant assez nettement comme déjà observé plus haut.

La ligne du bas de la Figure 6 montre l’intensité des rotations dans des coupes verticales de l’échantillon complet, aux différents incréments. Les incréments 1 et 3 confirment les observations faites sur les sous-volumes, à savoir absence d’organisation dans le premier incrément, et forte concentration dans la bande dans l’incrément final. Cependant l’incrément 2, situé au pic, montre que les rotations sont organisées à l’échelle de l’échantillon, dans une bande large d’une quinzaine de grains, centrée sur ce qui deviendra la bande de cisaillement développée, nettement plus étroite, dans les incréments suivants. Il faut noter que l’observation des sous-volumes seulement ne permet pas de détecter cette phase d’organisation globale précoce, car elle se déroule à l’échelle de l’échantillon.

4 CONCLUSION ET PERSPECTIVES

Les développements récents de la micro-tomographie à rayons X ont rendu possible d’imager en 3D de manière non destructive et non invasive de

olutions qui étaient tout simplement inimaginables voici vingt ans. Dans le contexte de la mécanique des sols, c’est un vieux rêve qui est devenu réalité : suivre individuellement tous les grains d’un échantillon en cours de déformation. La combinaison de cette nouv

oir-faire expérimental permettant de mener à bien des essais mécaniques représentatifs sous faisceau RX d’une part, et un savoir-faire numérique en analyse par corrélation d’image d’autre part, est extrêmement puissante. Nous sommes capables aujourd’hui d’extraire du volume énorme de données produites par ces essais, de nouveaux éléments d’analyse des mécanismes de déformation à l’œuvre au cœur des échantillons, tout au long d’un essai. On peut ainsi observer l’évolution de c

la résistance mobilisée, puis pendant que la déformation diffuse se développe, puis se localise et conduit finalement à un plateau caractérisé par une résistance résiduelle dégradée par rapport à la résistance de pic.

Un projet long terme qui découle naturellement de ces observations consiste à étudier l’évolution des contacts et leur rôle dans les mécanismes observés. Des travaux dans c

3D est désormais possible. D’un point de vue théorique, l’étude des bandes de

jouer à la loi de comporterecherche d’une discondéformation. Cependant, c’est seulement avec des approches en milieux enrichis que la notion d’épaisseur de bande de cisaillement émerge des équations. Les données expérimentales produites ici sont pertinentes pour établir de telles approches.

Du point de vue de l’application, les résultats concernant le développement des bandes de cisaillement en 3

grains, sont impordéformation affecte qu’excavations et pentes naturelles ou artificielles, digues et

rages, tunnels et galeries, forages, sites de stockage.

arque : une version en couleur de ce texte est disponible : ://l3sphnum.hmg.inpg.fr/HomepageS1/etagere/ICSMGE_Ando.pdf

REMERCIEMENTS

Ces études ont été rendues possibles par le financement attribué par l’ANR « blanche » à deux projets successifs, MicroMODEX

05-2008) et GeoBridge (2009-2013) qui ont permis de créer stallation de tomographie et de rassembler ou développer

autour d’elle les compétences et outils nécessaires.

RÉFÉRENCES

Alonso-Marroquin, F., and Vardoulakis, I. (2005) Micromechanics of shear bands in granular media. Powders and Grains, 701–704. ò, E., Hall, S.A., Viggiani, G., Desrues, J., and Bésuelle, P. (2012a) Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach. Acta Geotechnica, 7(1), 1-13.

Andò, E., Hall, S.A., Viggiani, G., Desrues, J., and Bésuelle, P. (2012b) Experimental micromechanics: grain-scale observation of sand deformation. Géotechnique Letters, http://www.icevirtuallibrary.com/content/serial/geolett.

vetti, F., Combe, G., and Lanier, J. (1997) Experimental micromechanical analysis of a 2D granular material: relation between structure evolution and loading path. M

Modelling crushing of granular materials as a poly-disperse mixture

Modélisation de la fracturation des matériaux granulaires comme un mélange poli-disperse

Caicedo B. Department of Civil and Environmental Engineering, Universidad de los Andes, Bogotá, Colombia

Ocampo M. Department of Civil Engineering, Universidad Javeriana, Bogotá Colombia

Vallejo L. Department of Civil and Environmental Engineering, University of Pittsburg

ABSTRACT: This paper presents a new model to assess the evolution of the grain size distribution of granular materials during loading. This model is based on the theory of poly-disperse mixtures proposed by De Larrard, 2000. Using this model it is possible to evaluate the compacity of the mixture depending on the grain size distribution, the shape of the particles and the compaction energy. Markov processes are used to assess the evolution of the grain size distribution, Markovian transition probabilities for each grain size are evaluated experimentally using gyratory compaction. Finally the experimental results are compared with the results of the modelshowing a very good agreement.

RÉSUMÉ : Cet article présente un nouveau modèle qui permet de calculer l’évolution de la granulométrie des matériaux granulairessous différents chargements. Le modèle est basé sur la théorie des mélanges poli-disperses proposée par De Larrard, 2000. Avec ce modèle, il est possible d’évaluer la compacité du mélange granulaire en fonction de la granulométrie du mélange, la forme desparticules et l’énergie de compactage. Un procédé de Markov est ajouté au modèle pour obtenir l’évolution de la granulométrie, lesprobabilités de transition étant évaluées expérimentalement à l’aide d’une machine de compactage giratoire. Finalement, les résultats du modèle sont comparés avec les résultats expérimentaux et montrent une très bonne correspondance.

KEYWORDS: granular materials, crushing, abrasion, poly-disperse mixtures, compaction.

1 INTRODUCTION

Particle fracture plays a major role in the behaviour of granular materials that are used in engineering structures such as paved roads, railroads, highway embankments, and foundations. The most important engineering properties of granular materials in these structures depend on the amount of particle crushing that occurs due to static or dynamic loads (Lade et al. 1996). Particle breakage occurs as a result of these loads (Bolton 1999; Feda 2002; Hagerty et al. 1993; Hardin 1985; Lade et al. 1996). Grain crushing is influenced by grain angularity, grain size, uniformity of gradation, particle strength, porosity, stress level, and anisotropy (Bolton and McDowell 1997; Feda 2002; Hagerty et al. 1993; Hardin 1985; Lade et al. 1996; Lobo-Guerrero 2006; McDowell and Bolton 1998; Nakata et al. 1999; Nakata et al. 2001a; Yamamuro and Lade 1996).

Researches carried out in the past 10 years have proved the capability of discrete element models to analyse crushing of granular materials. These models works with individual particles and permits stress analysis in each one of those particles. However such models require restrictive assumptions with respect to the number and shape of each particle and a better approach to the actual state of the granular material requires high computational cost. The model presented in this paper use the theory of poly-disperse materials proposed by De Larrard (2000) and is a new possibility to assess the evolution of grain size distribution of granular materials without dealing with the difficulties of discrete element modelling.

2 DESCRIPTION OF THE MODEL

The evolution of the grain size distribution of granular materials under cyclic loading is the result of the abrasion and crushing of particles. The following aspects must evaluate to assess such evolution: (i) the stress level in each class of particle size within the granular material; (ii) the strength of the particles taking in consideration the number of loadings for cyclic loading; and

(iii) the change of grain size distribution as a result of the crushing of particles.

In this model the first aspect is assessed with the aid of the poly-disperse theory proposed by De Larrard (2000), the second aspect uses the Weibull theory including a fatigue law for the cyclic loading, and the third one uses the theory of Markovian processes.

2.1 Description of a granular material as a polidisperse mixture

One of the most important variables having an effect on the crushing of a granular mixture is their unit weight. In fact huge experimental and theoretical evidence show that a large volume of voids in the granular material increases the stress within particles. As usual, the void volume could be characterized by the porosity n, the void ratio e or the compacity. The compacity of a granular mixture is defined as the solids volume of the grains in a unit volume. As a result the compacity could be calculated using the porosity or the void ratio as follows:

1n 11

e (1a, b)

The model proposed by De Larrard (2000) permits obtaining the compacity of a granular mixture depending on the volumetric proportion (yi) of particles of size di. This calculation is based on the knowledge of the residual compacity i of each grain size that represent the maximum compacity obtained experimentally for each class individually.

The virtual compacity was defined by De Larrard as the maximum compacity theoretically reachable for a particular granular mixture without any alteration of the original shape of the particles. Details about the derivation of the relationships to obtain the virtual compacity of a poly-disperse mixture are

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presented in De Larrard (2000), this paper present only a succinct description.

2.1.1 Compacity of granular mixtures For binary mixtures with grain size d1 and d2 three cases could be considered depending upon the interaction between the two types of grains: no interaction, full interaction and partial interaction. Figures 1(a) and 1(b) illustrates the case of a binary mixture with no interaction between particles. For this case the virtual compacity could be obtained knowing the volumetric concentration, y1 and y2, and the residual compacity, 1 and 2,of each class size as follows:

The total volume is: (2) 121 yy

(a) (b) Figure 1. Case of binary mixtures with no interaction.

The partial volume of each class is the volume of the class in the unit volume, then:

21

11

y and 21

22

y (3a, b)

The virtual compacity for a binary mixture becomes:

21 (4)

When the bigger grains are dominant, the virtual compacity is 1. In this case, the bigger grains fill up the volume with no interaction with the small grains. For this reason the partial volume 1 is the same than the residual compacity 1, so

11 . Then accordingly with equations 2 to 4, the virtual compacity 1 becomes:

)1( 211 y (5)

When the smaller grains are dominant the virtual compacity is 2. In this case the smaller grains fill the voids existing between the bigger grains with their maximum individual compacity 2. In this case the virtual compacity 2 is:

1222 11 y (6)

For the binary mixture only one virtual compacity is possible, this compacity is the minimum between 1 and 2. In fact if is higher than 1 grains 2 penetrate into grains 1 and vice versa. For this reason the only possible arrangement correspond to the minimum virtual compacity. This condition is called the impenetrability condition:

21,inf (7)

Different interactions between particles can be considered: a binary mixture having total interaction occurs when the size of the particles is identical but the residual compacities are different: 21 ;dd 21 . On the other hand, for the case of binary mixtures with d1>d2 two physical effects could appear: de-compaction effect and boundary effect. Taking in to account

these two effects, De Larrard (2000), calculate the virtual compacity 1 and 2 as follows:

22112

11

11 ya

(8)

112212

22

1111 yb

(9)

Where a12 is the de-compaction coefficient and b12 is the boundary effect coefficient.

In the case of a poly-disperse mixture with nc granular classes and with d1>di>dnc, the grains d>di in the mixture undergo a de-compaction effect due to the grains which size d<di and the mixture undergo a boundary effect due to the grains which size is d<di. The virtual compacity, considering the grain size i as the dominant grains is, (De Larrard, 2000):

1

1 111111

i

j

n

ijj

j

iijj

jiiji

ii

yayb

(9)

Similar than the case of binary mixtures, the impenetrability restriction is applicable. This restriction becomes:

ini inf

1 (10)

Once calculating i considering each class i as the dominant class (using equation 10), the actual dominant grain size is the one for which the minimum i is obtained. From the geotechnical point of view, all the grains with d<di are the matrix of the mixture, and the grains with d>di are dispersed grains in the mixture. The virtual compacity is unreachable experimentally. For this reason it is necessary to obtain the actual compacity, , which is more or less close to the virtual compacity depending upon the compaction method (<). For a real mixture the compacity is the accumulation of the compacitys corresponding to each class:

n

ii

1 (11)

In the mixture the dominant grain has the maximum compacity, taking in to account the presence of the other grains, this compacity is i .Therefore the compacities in the mixture are: .... , , ....... .

*

* 0 1i i 1i nTo obtain the relationship between the virtual compacity and

the actual compacity, De Larrard, 2000, proposes a compaction coefficient K. This compaction coefficient is the addition of the compaction coefficient corresponding to each grain size:

n

iiKK

1 (12)

The compaction coefficient for each grain size is obtained as follows:

i

ii

ii

iii

yK

111 *

*

(13)

2.2 Probability of crushing of particles depending on its compacity

The relationship i/i* is a powerful parameter to assess the

stress level supported by the particles of size di within the granular material. In fact, i is the volume filled by the

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particles di, and i* is the maximum volume that the particles di

can fill without any alteration of the mixture. Therefore the relationship i/i

* describes the proportion of voids that is filled by particles i with respect to the available void space for these particles (Figure 2).

Figure 2. Schematic drawing of the relationship i/i*.

Regarding stresses, a relationship i/i* close to 1 indicates

that particles i are filling well the void space reserved for those particles and therefore it is highly probably that the stress chains within the granular material flow along these particles. On the other hand, a relationship i/i

* close to zero indicates that particles i are in a loose state within the granular material and therefore doesn’t support significant level of stress.

Regarding crushing, the relationship i/i* take action with

two opposite trends: (i) when i/i* is close to 1 particles take

high level of stresses that increase the probability of crushing but these particles are well confined by other particles that reduces the probability of crushing; on the other hand when i/i

* is close to zero, particles are with low level of stress but have a reduced number of contact points with other particles. These two opposite trends suggest that there is a particular value of i/i

* for which the probability of crushing is maximum, this can be set as the compacity relationship for maximum crushing (i/i

* )mc.

Figure 3. Probability of crushing of particles depending on the relationship i/i

*.

A function that can describe the probability of crushing of particles depending the on the relationship i/i

* is presented in equation 14 (Ocampo 2009), Figure 3:

** 14)( iiiif ip (14)

Parameter is directly related with the compacity relationship for maximum crushing (i/i

* )mc as follows:

mcii )ln()2ln( * (15)

2.3 Probability of crushing of particles depending on its strength

The effect of the particles size on the strength of particles can be captured using the Weibull relationship. Furthermore as the

model presented in this paper focuses on cyclic loading, the strength of the particles depends on the number of loading cycles through a Wohler law. Equation 16 represent the crushing probability of any particle that is the result of including the Wohler law into the Weibull relationship:

mb

ib

ifcf

idNdp )(exp1)( 1 (16)

Where m is the Weibull Law parameter, is the applied stress, 1 is the strength of a particle having unit diameter and for one loading cycle, N is the number of cycles, bf is the slope of the Wohler fatigue law, and bc is the fracture parameter proposed by Lee (1992).

2.4 Combined probability of crushing

The crushing probabilities depending on the compacity and on the strength correspond to independent process; as a result the combined probability of crushing can be assessed as the product of both probabilities:

)()( ifff dpppi

(17)

2.5 Grain size distribution of crushed particles

When original particles break, they produce a set of smaller sub particles modifying the grain size distribution of the mixture. Afterwards, for another loading cycle, these sub particles can break again. This process is repetitive for different loading cycles and can be described by a Markov process. In such type of process an initial particle having di size (state 1) breaks and produce a set of particles having sizes dj<di (states 2 to 11), this process is described in Figure 4.

Figure 4. Diagram of state transition in a Markovian process.

In a Markovian process the transition between states is described by the Transition probability Matrix as follows:

nn

niii

n

ppppp

,

,,

,11,1

000

. (18)

In this matrix components pii are the probability of no failure of particle i that is obtained from equation 17 as pii=1-pfi.Elements pij for j<i represent the transition probability of a particle having an initial size i to a size j. Transition probabilities of particles during crushing was studied experimentally by Ocampo 2009 using particles with different colour for each initial size. Figure 5 represent the fitting of the experimental results using a beta function.

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Figure 5. Experimental probability transition of a particle di in to a particle dj.

Figure 7. Evolution of the grain size distribution for different number of loading cycles.

4 CONCLUSION Elements pij of the transition probability matrix can be

calculated as follows:

),( jifij ddBetapp (19)

Finally the evolution of the grain size distribution of a granular mixture results from the product of the transition probability matrix (transposed) and the grain size distribution before the loading cycle:

This paper presents a new model to assess the process of changes of the granular material properties as fracturing and abrasion occur as a result of cyclic loading. The model has shown that it can accurately predict the deterioration process of unbound granular materials subject to cyclic loading. This analytical model is based on the theory of poly-disperse mictures and therefore calculations up to high number of loading cycles can be performed without the difficulties of discrete element modelling. The results show very good agrement with the experimental tests illustrating the possibilities of this new model. 1 Ni

TNi yy (20)

5 REFERENCES3 RESULTS

To verify the predictions of the model three different granular materials were tested in a gyratory compactor. This apparatus reproduce the stress rotation during field compaction of granular layers. Compaction was performed to different number of loading cycles and then the grain size distribution was analyzed.

Bolton M D (1999). The role of micro-mechanics in soil mechanics Proceedings of the international workshop on soil crushability. Yamaguchi, Japan, 58-82

Bolton M D, McDowell G R Clastic mechanics IUTAM Symposium on Mechanics of Granular and Porous Materials. Cambridge, 35-46

De Larrard F (2000) Compacite et homogeneite des melanges granulaires. In: L. C. d. P. e. Chaussées (ed) Structures Granulaires et Formulation des Betons, 1st edn. LCPC.

Feda J (2002) Notes on the effect of grain crushing on the granular soil behaviour. Engineering Geology, 63(1-2): 93-98

Hagerty M M, Hite D R, Ullrich C R, Hagerty D J (1993) One-dimensional high-pressure com-pression of granular media. Journal of Geotechnical Engineering, 199(1): 1-18

Hardin B O (1985) Crushing of soil particles. Journal of Geotechnical Engineering, 111(10): 1177-1192

Lade P V, Yamamuro J A, Bopp P A (1996) Significance of Particle Crushing in Granular Ma-terials. Journal of Geotechnical Engineering 122(4): 309-316

Lee, D. M. (1992). "The angles of friction of granular fills," Ph.D. dissertation, University of Cambridge.

Lobo-Guerrero S (2006) Evaluation of crushing in granular materials using the discrete element method and fractal theory. University of Pittsburgh, Pittsburgh, PA.

McDowell G R, Bolton M D (1998) On the micromechanics of crushable aggregates. Géotechnique, 48(5): 667-679

Nakata Y, Hyde F L, Hyodo M, Murata H (1999) A probabilistic approach to sand particle crushing in the triaxial test. Géotechnique, 49(5): 567-583 Figure 6. Comparison between experimental and model results.

Figure 6 present a comparison of the experimental and numerical tests for one of the tested materials. This figure shows the mass of particles retained in each sieve size. Each point in the figures corresponds to a particle size at the end of a specific number of loading cycles. These figures show the very good agreement between the model and the experimental data.

Nakata Y, Hyodo M, Hyde F L, Kato Y, Murata H (2001) Microscopic particle crushing of sand subjected to high pressure one-dimensional compresion. soils and Foundations, 41(1): 69-82

Ocampo M (2009) Fracturamiento de partículas en materiales granulares sometidos a cargas cíclicas con rotación de esfuerzos. Universidad de Los Andes, Bogotá D.C.

Figure 7 represent an extrapolation of the results of the model up to a million of loading cycles showing the capacity of this analytical model to calculate the evolution of grain size distribution when a high number of cycles are applied to the material.

Yamamuro J A, Lade P V (1996) Drained Sand Behavior in Axisymmetric Tests at High Pres-sures. Journal of Geotechnical Engineering, 122(2): 109-119

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Behaviour of a compacted silty sand under constant water content shearing

Comportement d'un sable limoneux compacté sous cisaillement à teneur en eau constante

Heitor A., Rujikiatkamjorn C., Indraratna B. Centre for Geomechanics and Railway Engineering, University of Wollongong, Wollongong, Australia ; ARC Centre of Excellence in Geotechnical Science and Engineering, Australia

ABSTRACT: The structure derived from compacting the soil at different water contents and energy levels can have a substantial effect on its shear strength. While the shear strength with varying suction can be estimated based on the saturated shear strengthparameters and the unsaturated angle of shearing resistance (b), limited studies have explored the variation of shear strengthproperties with different compaction states. In this paper, the shear strength of a silty sand soil was investigated using a conventionaldirect shear box under constant water content condition. The tests were conducted on specimens prepared by Proctor compaction withthree different normal pressures. The shear strength parameters were obtained and modelled in terms of ultimate states.

RÉSUMÉ : La structure interne d’un sol résultante du compactage à différentes teneurs en eau et niveaux d'énergie peut avoir un effet important sur sa résistance au cisaillement. Alors que la résistance au cisaillement avec succion variable peut être estimée sur la basedes paramètres de résistance au cisaillement en conditions saturées et de l'angle de résistance au cisaillement non saturé (b), peu d'études ont exploré la variation des propriétés de résistance au cisaillement avec des niveaux de compactage différents. Dans cetarticle, la résistance au cisaillement d'un sol de sable limoneux a été étudiée en utilisant une boîte de cisaillement direct classique. Les tests ont été effectués sur des échantillons préparés par la méthode de compactage Proctor avec trois différentes pressions normales. Les paramètres de résistance au cisaillement ont été obtenus et modélisés en termes d’états ultimes.

KEYWORDS: Compacted soil, shear strength, constant water content, direct shear test.

1 INTRODUCTION.

Field compaction may not always be uniform (i.e. differences in hydration time, lift thickness and soil variability). Early studies have shown that these variations can have substantial effects on the mechanical behaviour of the compacted fine-grained soils (Proctor, 1933; Seed and Chan, 1956). The main aspects that can be affected are related to swelling, wetting induced collapse compression and soil response due to external loading (compression behaviour and shear strength).

Research studies on the shear strength behaviour of unsaturated compacted soil prepared at different compaction states (i.e. water contents and energy levels) showed that there is an intimate relationship between shear strength and water retention properties (i.e. Vanapalli et al. 1996). Furthermore, Wheeler and Sivakumar (2000) reported that a change in water content during compaction produces variations in the positions of the normal compression and critical state lines. Toll and Ong (2003) modelled the ultimate (critical) shearing behaviour of soil prepared at different initial compaction states and introduced the critical stress ratios as a function of the degree of saturation (Sr). Tarantino and Tombolato (2005) investigated the shear strength and hydraulic behaviour of statically compacted kaolin and showed that some of stress-strain behaviour features observed can only be modelled using hydro-mechanical coupling models.

While there has been an intensive research effort dedicated to the study of shear strength properties with varying post-compaction suction, limited research studies focussed on investigating the shear strength properties for different compaction states. This is undoubtedly important considering the lack of field compaction uniformity in terms of water content and energy. This paper presents the results from direct shear tests performed on compacted silty sand. The specimens were prepared at different compaction states in order to explore a broad range of initial conditions. Tests were carried out using

a conventional direct shear box with the adoption of special measures to maintain constant water content conditions.

1.1 Constant water content direct shear tests (CWDST)

There are a number of different types of apparatus that can be used to study the shear strength behaviour of unsaturated soils. For testing soil under unsaturated conditions, the conventional apparatus often needs to be modified to enable the suction to be controlled or measured during the shearing stages, using i.e. axis translation technique, vapour equilibrium or osmotic suction control. While these types of control are effective, the laboratory conditions may not always be truly representative of those in the field, where shearing typically occurs under constant water content conditions. The use of the conventional direct shear box for determining the unsaturated shear strength parameters is very attractive because it is readily available to practitioners. Although it requires a careful moisture control, it can benefit from higher rates of shearing (compare 1m/min for suction controlled apparatus with 0.0051mm/min, i.e. Zhan and Ng 2006, Oloo and Fredlund 1996). The only drawback is the absence of an independent system to measure suction, although, Oloo and Fredlund (1996) and Cokca et al. (2004) assumed that any changes in suction during shearing would be minimal provided that a relatively fast rate of strain is adopted.

2 EXPERIMENTAL WORK

2.1 Soil type

The soil used in this study was silty sand classified as SP-SC (Unified Soil Classification System, USCS). The soil is a by-product of cobble quarrying activities that has been widely used to fill low areas at the Penrith Lakes site in Penrith (NSW, Australia). While the soils present on site are quite variable, for

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this study only a single grading was used. The particle size distribution was composed of 89% sand and 11% fines, of which 7% is silt and the remaining 4% is clay size particles. It has a liquid limit of 25.5%, a plasticity index of 10 and specific gravity of 2.7.

2.2 Laboratory testing program

The laboratory testing program included the execution of Proctor compaction tests (AS1289.5.1.1, 2003) under different levels of compaction energy (i.e. 15, 25, 35 blows per layer corresponding to 358, 596 and 834kJ/m3). The required amount of water was added to the sample and the mixture was thoroughly mixed with a masonry trowel and then left under constant temperature and humidity conditions for 24h to ensure a uniform distribution of moisture. The compaction data is shown Figure 1. Subsequently, the specimens were carefully trimmed (606025mm3) from the compacted soil cylinders (1L) to minimize disturbance, while the excess of soil was typically used to determine the water content and suction using filter paper method and a small tip tensiometer. The procedure was completed in a matter of minutes to minimize exposure to air to prevent the loss of any moisture. Thereafter the specimens were wrapped in cling film and stored inside a plastic bag.

6 8 10 12 14 16 1815

16

17

18

19

20

21

Sr=0.67

Reduced, 15bl/layer : E=358kJ/m3

Standard, 25bl/layer : E=596kJ/m3

Enhanced, 35bl/layer : E=834kJ/m3Dry

uni

t wei

ght, d (k

N/m

3 )

Water content, w (%)

Sr = 1

Sr=0.8

CWDST tests

Figure 1. Compaction curves obtained for the silty sand soil.

2.2.1 CWDST program A conventional shear box apparatus with a carriage running on roller bearings and a step motor drive unit capable of applying constant rate horizontal displacements was used. The apparatus was equipped with a load cell and two LVDT displacement transducers for measuring the horizontal shear force and monitor the horizontal and vertical displacements (accuracy of 0.002kN, 0.0025mm and 0.001mm, respectively). A vertical load was applied with a lever arm loading system with beam ratio of 10:1. Data acquisition was controlled by a LabVIEW program coded “in house” accompanied with a National Instruments card NI USB-6009 with 8 input channels.

To conduct the tests under CW conditions, an effort was made to prevent any evaporation. This was achieved by running the compression and shearing stages of the tests in a temperature controlled environment (232oC), and by enclosing the direct shear box with the assembly in an air tight plastic bag (Figure 2). Furthermore, to minimize the volume of air around the specimen, a 1mm thick film of polyethylene was placed in contact with the specimen and the gaps between the two sliding halves and the bottom half and base were sealed with silicone grease. The compacted specimens were extruded into the shear box and then subjected to a compression stage (vertical stresses of 38.4kPa, 79.5kPa, and 146.7kPa). Subsequently, the specimens were sheared at a constant rate of displacement of 0.01mm/min. It is important to note that in a direct shear test, the shear zone is localised and thin compared to the mass of the specimen (Shibuya et al. 1997). While suction is likely to be constant throughout the specimen when a small displacement rate is adopted due to self-equilibration, water content likely differs, but on average it would the same as the initial water content because water is not allowed to flow out and

evaporation is minimised. This rationale is supported by the slight difference in suction measured using filter paper method (ASTM D5298, 2003) at the beginning and end of the tests (<5kPa and <1kPa for specimens prepared at dry and wet of OMC, respectively) and a small vertical variation of water content typically less 0.2-0.3% obtained in the sheared specimen at the end of the test.

Figure 2. Shear box diagram with the system implemented to prevent evaporation.

3 RESULTS AND DISCUSSION

3.1 As-compacted water content and suction

Figure 3 shows the water retention data for additional compacted specimens prepared at different energy levels. Overall, the suction decreases with an increasing water content varying between 5 kPa to 616 kPa. Although there is no apparent relationship between suction and compaction energy, all data points seem to converge to a logarithmic regression line given by Eq. (1) (R2 > 0.95).

( ) -1.56ln(s)+18.50w s (1)

This indicates that the hydraulic behaviour of compacted soil may be independent of the compaction characteristics (i.e. change in the water content and energy level).

1 10 100

8

10

12

14

16

18

1000

Moi

stur

e co

nten

t, w

(%)

Matric suction, s (kPa)

R2>0.95-1.56ln(s)+18.50

Compaction energy: E2=358 kN.m/m3

E3=596 kN.m/m3

E4=834 kN.m/m3

Figure 3. Post-compaction matric suction data in terms of water content.

3.2 Shear strength behaviour

In general, all the specimens showed a peak followed by a decrease in shear stress before subsequently attaining a relatively constant value after 6.5 mm of horizontal displacement. Although strain softening and dilative behaviour was clearly evident in the specimens compacted at dry of OMC, the specimens compacted at OMC and wetter of OMC did not show a distinct post peak drop and showed a mainly contractive response. Figure 4 shows the typical displacement plots obtained for an applied vertical stress of 40 kPa.

3.2.1 Peak and ultimate shear strength envelopes

Figure 5(a) shows the failure envelopes for the specimens prepared at same energy level but different water contents grouped by the correspondent applied vertical stresses. Both peak and ultimate states are represented and the envelopes were interpolated using the procedure suggested by Vilar (2006). While both peak and ultimate shear strength increase with

1009


suction in a non-linear fashion, the rate of increase seems to gradually decrease with increasing suction. Moreover, the envelopes seem to suggest the existence of a critical suction value, after which shear strength increase with suction is not very significant. For this particular study this value seems to be at suction of 100kPa, which corresponds to approximately 11.3% water content. It is interesting to note that the difference between the peak and ultimate shear strength envelopes increases with the applied vertical stress. This difference is probably due to the larger reduction in void ratio attained at the end of the compression stage for the specimens tested with higher vertical stresses. At the beginning of the test, these specimens have denser particle arrangement and would likely experience a more pronounced softening behaviour that in turn causes larger differences between the peak and ultimate shear strength. In figure 5 (b) the peak and ultimate shear strength data of specimens prepared at approximately the same water content is represented with the level of compaction energy. The peak shear strength seems to decrease with increasing energy while ultimate shear strength is less affected. This difference may be attributed to the fact that initial soil structure is being erased during shearing. The differences in peak shear strength are then probably associated with the difference in soil structure, particularly when the line of optima is exceeded (Figure 1, i.e. Kodikara, 2012). In addition, the specimens for whom the compaction end states are located on the wet side of the compaction plane may have experienced during compaction larger pore water pressures that were quickly dissipated. This in conjunction with the change in structure may contribute to the deterioration of the soil strength; however, further confirmation of this hypothesis is desirable.

0 2 4 6 80

20

40

60

80

100

10

Shea

r stre

ss,

(kPa

)

Horizontal displacement, x (mm)

(a)

0 2 4 6 8-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

10

w = 8.5%w = 10.5%w = 12.8%w = 12.6% (356kJ/m3)w = 13.0% (834kJ/m3)w = 14.0% (834kJ/m3)w = 14.1% w = 16.8%

Ver

tical

dis

plac

emen

t, y

(mm

)


(b)

Figure 4. Shear tests results for an applied vertical stress of 38.4kpa in terms of (a) shear stress and (b) vertical displacement.(w =8.516.8% and E=358, 596 and 834kJ/m3)

3.2.2 Ultimate shear states

The ultimate shear state results were modelled using two different approaches, namely, (a) the average skeleton stress (Tarantino and Tombolato, 2005) and (b) critical stress ratios (Toll and Ong 2003). Both approaches make reference to the saturated states which were found to be relatively independent of the compaction characteristics (Figure 6). Saturated tests were only conducted for specimens compacted at 12.5%, given the similarities of stress-strain behaviour between the saturated

specimens and as-compacted specimens that reached saturation conditions during compression and shearing.

The average skeleton approach is based on the assumption that the water menisci have a negligible effect on the ultimate shear strength. The shear strength, , is given by considering the shear strength of saturated states at the same average skeleton stress sat and the degree of saturation of the macropores, or Srm, as follows:

w wmsat v rm sat v

wm

e esS se e

(3)

where ew and e are the water ratio (ew=eSr) and void ratio, respectively, and ewm is the microstructural water ratio. The value of ewm adopted is 0.237 and it was found by the least squares method fitting of Eq. (3). Figure 7 (a) shows the comparison between the measured and predicted ultimate shear strength for all specimens, considering the average skeleton stress defined in terms of Sr and Srm. The prediction of shear stress is favoured by the adoption of the Srm instead of Sr.Similar observations were reported by Tarantino and Tombolato (2005) for statically compacted kaolin, despite the fact that the fabric considered was mainly representative of the dry side of optimum.

0 100 200 300 400 500 600 7000

25

50

75

100

125

150

175(a)

, v=38.4kPa

, v=79.5kPa

, v=146.7kPa

Peak shear strength envelopes Ultimate shear strength envelopes

Shea

r stre

ss,

(kPa

)

Matric suction, s (kPa)

300 400 500 600 700 800 9000

25

50

75

100

125

150

175(b)

Shea

r stre

ss,

(kPa

)

Compaction energy, E (kJ/m3)Figure 5. Shear strength envelopes for specimens compacted at (a)

energy level of 596kJ/m3 and (b) w=12.813% (close and open symbols represent peak and ultimate states, respectively).

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

10

AC-14.1(38.4)

AC 14.0(38.4)

AC-14.1(79.5)

AC-13.0(146.7)

AC-12.8(146.7)

sat-12.5(146.7)

sat-12.5(79.5)sat-12.5(38.4)

Stre

ss ra

tio , v


AC: As-compacted CWDSTNumbers in brackets represent vertical pressures

Figure 6. Stress ratio with horizontal displacement for saturated drained tests (wc=12.5%, solid symbols) and constant water content test at high Sr (open symbols).

In contrast, in the critical stress ratio approach the contribution of the total stress and suction to the shear strength is considered separately (Toll and Ong, 2003). The soil fabric changes are reflected in the variation of two individual stress ratios governed by the degree of saturation. Toll and Ong

1010


(2003) proposed the general shear strength relationship for critical state as follows

tan tana bv s

(4)

where, a , b, are the critical state friction angles in respect to the vertical stress, v and suction, s. The critical state friction angles can be found using the following functions:

2

1 2'

kbbr r

r r

S SS S

(5)

2

1 2max max

1' ' '

kaa a ar r

r r

S SS S

(6)

where Sr1 and Sr2 are two reference degrees of saturation and ’ is the saturated friction angle. For Sr exceeding Sr1, a/’ and b/’ ratios are equal to ’ whereas for Sr smaller than Sr2, a/’=(a/’)max and b/’=0. In this study the reference degree of saturation was taken as Sr1=1 (full saturation conditions) and Sr2=0.75, kb=2 and ka=1 (Figure 7 b).

Figure 7. Prediction of (a) the ultimate shear strength using the average skeleton approach and (b) the critical friction angles with Sr measured at critical state.

The predictions of the ultimate shear strength obtained with the two approaches are shown in Figure 8. Although both approaches require approximately the same number of parameters, the critical friction angles approach seems superior in predicting the shear strength of soil prepared at a wide range of moisture contents and energy levels. The predictions may be considered satisfactory given that in this study a single set of parameters is used for modelling compaction states that differ in both water content and energy level and thus soil structure.

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

10kPa1:1

Measured shear stress, (kPa)

Pred

icte

d sh

ear s

tress

, (k

Pa) by Average skeleton approach

by Critical friction angles approach

Figure 8. Prediction of the ultimate shear strength using the average skeleton approach and using the critical friction angles approach

4 CONCLUSION

This study presented the results on the shear strength of compacted silty sand using constant water content direct shear tests. The as-compacted water retention data showed that, regardless of the energy level applied during compaction, there is a unique relationship between water content and suction.

The peak and ultimate shear strength envelopes results show that there is a relatively well defined non-linear relationship between shear strength and matric suction. Also, the envelopes suggest the existence of critical suction, after which the shear strength increase is less significant. A decrease of peak shear strength was observed for increasing compaction energy that

was interpreted to be associated with a difference in soil structure and the compaction history of the specimens.

Constant water content direct shear tests on saturated specimens show that the ultimate shear strength is relatively independent of the initial compaction state. The ultimate shear strength is modelled using two different approaches, that is, the average skeleton stress and independent critical stress ratios. The first is usually associated with the consideration of a Bishop type of effective stress whereas the latter considers the net stress and suction effect on the shear strength to be independent. Although both approaches provide reasonable estimations of shear strength, for this particular study the independent stress variables approach is superior. Worth noting that the prediction exercise catered for different compacted states and represents a step forward in understanding the implications of the inherent variability of compaction conditions on the soil shear strength.

5 ACKNOWLEDGEMENTS

The authors acknowledge the financial assistance provided by the Australia Research Council, Penrith Lakes Development Corporation, Ltd and Coffey Geotechnics and assistance from Mr. Robert Golaszewski and Mr. Alan Grant.

6 REFERENCES

Cokca, E., Erol, O. and Armangil, F. (2004). Effects of compaction moisture content on the shear strength of an unsaturated clay. Geotechnical and Geological Engineering 22(2), 285-297.

Jotisankasa, A. and Mairaing, W. (2010). Suction-Monitored Direct Shear Testing of Residual Soils from Landslide-Prone Areas. Journal of Geotechnical and Geoenvironmental Engineering136(3), 533.

Kodikara, J. (2012) New framework for volumetric constitutive behaviour of compacted unsaturated soils, Can. Geotech. J. 49, 1227-1243.

Oloo, S. Y. and Fredlund, D. G. (1996). A method for determination of fb for statically compacted soils. Can. Geotech. J. 33(2), 272-280.

Proctor, R. R. (1933). Fundamental Principles of Soil Compaction. Engineering News Record 111(9), 245-248.

Seed, B. and Chan, C. K. (1959). Compacted clays: Structure and strength characteristics. Journal of soil mechanics and Foundations division Transactions 126(I), 1344.

Shibuya, S., Mitachi, T. and Tamate, S. (1997). Interpretation of direct shear box testing of sands as quasi-simple shear. Géotechnique47(4), 769-790.

Tarantino, A. and Tombolato, S. (2005). Coupling of hydraulic and mechanical behaviour in unsaturated compacted clay. Géotechnique55(4), 307-317.

Toll, D. G. and Ong, B. H. (2003). Critical-state parameters for an unsaturated residual sandy clay. Géotechnique 53(1), 93-103.

Vanapalli, S. K., Fredlund, D. G. and Pufahl, D. E. (1996). The Relationship Between the Soil-Water Characteristic Curve and the Unsaturated Shear Strength of a Compacted Glacial Till. Geotechnical Testing Journal 19 (3), 259-268.

Vilar, O. M. (2006). A simplified procedure to estimate the shear strength envelope of unsaturated soils. Can. Geotech. J. 43(10), 1088-1088.

Wheeler, S. J. and Sivakumar, V. (2000). Influence of compaction procedure on the mechanical behaviour of an unsaturated compacted clay. Part 2: Shearing and constitutive modelling. Géotechnique 50(4), 369-376.

Zhan, T. L. and Ng, C. W. W. (2006). Shear strength characteristics of an unsaturated expansive clay. Can. Geotech. J. 43(7), 751-751.

00

20 40 60 80 100 120 140 160 180 200

20

40

60

80

100

120

140

160

180

200

(a)

Measured shear stress, m (kPa)

Pred

icte

d sh

ear s

tress

, p (

kPa)

v (Srm)

v (S

r)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

10

20

30

40

50 (b)a

b

Crit

ical

fric

tion

angl

es,

(o )

Average degree of saturation, Sr

Sr2

a')max '

1011


1

SHEAR STRENGTH AND DEFORMATION OF METHANE HYDRATE BEARING SAND WITH FINES

Résistance au cisaillement et déformation des sables avec des fines contenant de l’hydrate de méthane

M. Hyodo, N. Yoshimoto & A. KatoYamaguchi University, Ube, Japan

J. YonedaThe National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan

ABSTRACT: A series of triaxial tests has been carried out to determine the mechanical properties and dissociation characteristics of sands with fines containing methane hydrate, using an innovative high pressure apparatus which has been developed to reproduce the in-situ conditions expected during proposed methane extraction methods. It was found that the strength of MH sand increased withMH saturation due to particle bonding and that the bonding effect was particularly dependent on the grain size of the host sand. Ahigh pressure and low temperature plane strain testing apparatus was also developed for visualizing the deformation of methanehydrate bearing sand due to methane hydrate production. Using this testing apparatus, plane strain compression and methane hydrate dissociation by depressurization tests were performed with the measurement of localized deformation.

RÉSUMÉ: Une série d'essais triaxiaux ont été effectués pour déterminer les propriétés mécaniques et les caractéristiques de dissociation des sables avec des fines contenant du hydrate de méthane (MH), en utilisant un appareil innovant de haute pression qui a été développé pour reproduire les conditions in-situ attendues pour des méthodes proposées d'extraction du méthane. Il a été constaté que la résistance du sable avec MH augmente avec la saturation du MH due au collage des particules et que l'effet de collageest particulièrement dépendant de la granulométrie du sable. Un appareil d'essai à haute pression et basse température en contraintesplanes a aussi été développé pour la visualisation de la déformation du sable contenant des hydrates de méthane grâce à la production de méthane hydraté. En utilisant cet appareil, des essais de compression en contraintes planes et de dissociation des hydrates de méthane par dépressurisation ont été effectués avec la mesure de la déformation locale.

KEYWORDS: metane hydrate, sand, fines, temperature, high stress triaxial test, high stress plane strain test, local deformation

1 INTRODUCTION

Recently there has been much research into Methane Hydrate (referred to as MH hereafter) in the deep seabed as a developable material. In Japan, an MH rich layer was found in the Nankai Trough and production tests will be performed from 2013 (MH21 Research Consortium, 2012). MH in the deep sea bed can exist at certain water pressure and temperature conditions. It exists in the pore space of the sand, bonding the sand particles. The MH rich layer is located around 100m-300m from the seabed, in deep seas with depths of over 1000m. As MH exists in uncemented sand sediments, there are many geotechnical-engineering related problems in order to confirm the stability of the production well and the grounds in its vicinity (Yamamoto. K. 2009). Hyodo, et al. (2008) developed temperature-controlled high-stress triaxial compression testing apparatus, and used this to perform a series of shear tests on not only undisturbed MH bearing sand samples from Nankai Trough, but also artificially produced MH in Toyoura sand to investigate the variation of shear strength due to cementation of MH. In this study, specimens of sand with fines were prepared to simulate the sediments in Nankai Trough and MH was produced with various degree of MH saturation in the specimens. A series of triaxial tests was performed and a newly developed high-stress plane strain shear testing apparatus with an observation window was used to investigate the global and local deformation of MH bearing sand.

2 MATERIALS USED IN EXPERIMENTS

Grain size distribution curves for samples from Nankai Trough and the simulation materials prepared in this study are shown in

Figure 1. The sediments in Nankai Trough`s seabed soil constitute turbidite and show stratified layers with wide grain distribution curves, with contents ranging from sand to clay. The grain size distribution for the MH rich layer in Nankai Trough is shown in grey; it is mostly sand with fines content. In order to simulate the grain size distribution and minerals of this layer, silica sand, kaolin and mica were mixed and four kinds of simulated sands Ta, Tb, Tc, Td were prepared as host sands. The fines content increases in order of Tb, Ta, Tc, Td and the mean particle size decreases in order of Ta, Tb, Tc, Td.

3 SHEAR CHARACTERISTICS OF MH BEARING SANDSBY TEMPERATURE-CONTROLLED HIGH-STRESSTRIAXIAL TESTING

Figure 1 Grain size distribution curves (Nankai trough and artificial samples)

Shear Strength and Deformation of Methane Hydrate Bearing Sand with Fines

Résistance au cisaillement et déformation des sables avec des fines contenant de l’hydrate de méthane

Hyodo M., Yoshimoto N., Kato A.Yanaguchi Universit, Ube, Japan

Yoneda J.The National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan

1012



2

Production of MH bearing sands and their shear tests were conducted using the temperature-controlled high-stress triaxial testing apparatus (Hyodo, et al.2008). For the simulation material, MH was formed in these host sands with 50%, 30% and 0% degree of MH saturation. From the results of density tests on natural cores from Nankai Trough, the porosities of the simulation materials were set as n=40% and 45%. In this paper, only the results of n=45% are presented. The method of formation of MH in the simulation materials is as follows. Firstly, an initial water content equivalent to the desired degree of MH saturation was prepared and then an unsaturated specimen, with diameter 30mm and height 60mm, was made using tamping to a given density. Next, the specimen was set in the cell of the apparatus and methane gas was injected into the specimen for 24hr at 10MPa pore water pressure, 13MPa cell pressure and at a temperature of 5℃ to form the MH. After stopping the methane gas injection and confirming the formation of MH, the specimen was filled with pure water in order to saturate it. A given effective stress was then applied and shear tests were performed under drained conditions with a rate of shear of 0.1%/min. After the shear test, the pore pressure was decreased to a level outside of the MH stability boundary in order to dissociate the MH which had been produced. The quantity of MH was measured and the degree of MH saturation was calculated by this amount of dissociated gas. The triaxial test results for host sands Ta, Tb, Tc and Td with an initial porosity of n =45%, under an effective confining stress of 1MPa, 3MPa and 5MPa are presented in Figure 2. In the figure,the initial stiffness and residual strength slightly decreases with increasing fines content, however there is no marked difference. Volumetric strain induced by shear appears on the contractive side and increases with increasing fines content and confining stress. Figure 3 shows the variation of the secant modulus at 1% axial strain against fines content for each material with each confining stress. It can be seen that there is a trend for the secant modulus to decrease with increasing fines content at all confining stresses. However, this is not true for Ta and Tb, due to the effect of their mean particle size. Fig.4(a)-(c) show the shear testing results for MH bearing sands by using Ta, Tb and Tc as host sands. Tests were performed at a porosity of n =45% with effective confining stress of 1MPa at various degrees of MH saturation. From the figures it can be observed that initial stiffness and peak residual strengths increased as the degree of MH saturation increased. Corresponding volumetric strain increases and behaves in a more dilative manner with increasing degrees of MH saturation. In Figure 5, the difference between the peak strength of MH bearing sand and the strengths of the host sands, at an axial strain corresponding to the peak value of MH bearing sand, are normalized by effective confining stress and then plotted against the degree of MH saturation. It can be observed that the strength increased rapidly when the degree of MH saturation passed 30%, and the rate of increase decreases in order of Ta, Tb and Tc. It is therefore obvious that the grain size distribution affects the cementation effect of MH.

4 SHEAR CHARACTERISTICS OF MH BEARING SANDSBY HIGH-STRESS PLANE STRAIN SHEAR TESTING

An overview of the testing equipment (Yoneda et al. 2011) is shown in Figure 6. This apparatus can control temperatures and pressures equivalent to an MH reservoir in deep seabeds. Additionally, observation windows are installed in front of and behind the specimen in order to allow the local deformation of the specimen during shear tests to be measured. The specimen is a cuboid with 80mm width, 60mm thickness and 160mm height. A 5mmｘ5mm mesh was drawn on the membrane in front of the observation window. The observation was performed using a digital camera(g), which took pictures according to a timer controlled by a remote system. An LED(h) was installed to brighten the pressure cell(i), which allowed the specimen(e) to

Figure 2 Variation of deviator stress and volumetric strainagainst axial strain for host sands

0 10 20 30 40 500

1

2

3Ta

Tb

TcTd

Sec

ant m

odul

us E

( (

MP

a)

Fines content Fc(%)

n=45%

c'=1MPa

c'=3MPa

c'=5MPa

Figure 4 Variation of deviator stress and volumetric strain against axial strain for methane hydrate bearing sands(b) Tb(a)Ta (c) Tc

0

1

2

3

4

5

6

0

-6

0 10 20 30 Vo

lum

etr

ic s

tra

in

v(%

)

Axial strain a(%)

De

via

tor

stre

ss q

(MP

a)

Tb

c'=1MPa

SMH=0% n=45.5%SMH=31.1% n=45.5%

SMH=47.4% n=45.3%

0

1

2

3

4

5

6

0

-6

0 10 20 30 Vo

lum

etr

ic s

tra

in

v(%

)

Axial strain a(%)

De

via

tor

stre

ss q

(MP

a)

Tc

c'=1MPa

SMH=0% n=45.0%

SMH=24.3% n=45.2%

SMH=58.6% n=44.9%

0

1

2

3

4

5

6

0

-6

0 10 20 30 Vo

lum

etr

ic s

tra

in

v(%

)

Axial strain a(%)

De

via

tor

stre

ss q

(MP

a)

Ta

c'=1MPa

SMH=0% n=45.1%SMH=24.6% n=45.5%

SMH=43.7% n=44.8%

Figure 3 Relationship between secant modulusand fines content

0

4

8

12

12

6

0

0 10 20 30

Vo

lum

etr

ic s

tra

in

v(%

)

Axial strain a(%)

De

via

tor

stre

ss q

(MP

a)

SMH=0% n=45% a=30%

Ta

Tb

TcTd

1013



3

Figure 5 Variation of normalized deviator stress difference against degree of MH saturation for Ta, Tb, Tc

a

a

b

c

d

e

j

kl

60mm80mm

160mm

FrontSidea: Pressure gauge g: Camerab: Syringe pump(Upper) h: LED lightc: Top cap i: Pressure celld: Pedestal j:Syringe pump(Lower) e: Specimen k:Thermocouple(60mm)f: Confining plate l: Thermocouple(30mm)

Outside Insidea

a

i

b

j

e

d

c

hgf f

Figure 6 Schematic overview of the high pressure plane strain testing apparatus

Figure 7 Variation of principal stress difference and volumetric strain against axial strain for methane hydrate bearing sands

Figure 8 Volumetric strain and maximum shearing strain contours at axial strain εa = 10% by PIV analysis

0

3

6

9

12

5

0

-5

0 3 6 9 12

Pri

nci

pa

l str

ess

diff

ere

nce

(

'1-

'3)/

2(M

Pa

)

Vo

lum

etr

ic s

tra

in

v(%

)

Axial strain a(%)

Temperature=5℃

c=3.0MPa

Tc SMH=0.0%

Tc SMH=41.3%

Toyoura sand SMH=48.0%

Toyoura sand SMH=0.0%

0 10 20 30 40 50 600.0

0.5

1.0

1.5

2.0

Nor

mal

ized

dev

iato

r st

ress

diffe

renc

e (q

MH-q

sand

)/

c'

Degree of MH saturation SMH(%)

TaTb

Tc

n=45% Ta

'c=1MPa 'c=3MPaTb

'c=1MPa 'c=3MPa 'c=5MPaTc

'c=1MPa 'c=3MPa

be observed. Local deformation analysis was performed by observing the cross-points of the mesh during shear tests and using this data in PIV analysis. Thermocouples(k,l) were installed at 60mm and 30mm from the bottom of the specimen in order to measure the variation of the temperature during the dissociation of MH. Tc and Toyoura sand were the materials used for comparison. Specimens were prepared with water contents equivalent to given degrees of MH saturation and tamped in 12 layers to give a porosity of n =45%. Formation of MH was performed using the same method as in the triaxial compression tests. The specimens were saturated by filling the pore water and consolidated at given effective confining stress and then subjected to shear tests under drained conditions. The speed of shear was 0.1%/min. Also, in order to understand the behavior during MH production, MH was dissociated by decreasing pore water pressure by 7MPa under constant cell pressure and observing the behavior. After finishing MH dissociation, the pore water pressure again rises and the behavior was also investigated. The rate of depressurization during dissociation of MH and repressurization was 0.5MPa/min. Figure 7 shows the shear test results for Tc and Toyoura sand with and without MH. A marked increase in the strength and stiffness due to the cementation of MH are observed in both specimens. The increments of strength and initial stiffness for Tc are lower than those of Toyoura sand. The corresponding volumetric strain exhibits a dilative manner for MH bearing Toyoura sand, whilst that of Tc shows a contractive manner during shear. Figure 8 shows the results of the PIV analysis during shear for the specimens. The upper part of the figure shows volumetric strain and the bottom part shows shear strain contour for specimens at 10% axial strain. On the left hand side there is Tc without MH, then Tc with MH and then Toyoura sand without and with MH. Clearer shear bands can be observed in Toyoura sand than in Tc, so the effect of fines on local deformation can be observed in this figure. The shear band for Toyoura sand with MH is clearer than that of Toyoura sand without MH, but there are no clear differences observed between Tc with and without MH. Next, in order to simulate the production of MH, pore water pressure was decreased whilst keeping constant initial shear stress, MH was dissociated and the deformation behavior of loss of cementation was investigated. Specimens which had cell pressure 10MPa, pore water pressure 7MPa and consolidated at effective stress 3MPa had their pore water pressure decreased by 7MPa. Stress and temperature conditions were set outside of the MH stability state line and MH was dissociated. Figures 9 and 10 show effective stress paths applied to the experiments. In these figures, only the results for Toyoura sand are shown. Failure envelopes for Toyoura sand with and without MH are shown by broken and solid line, respectively. The dissociation tests were performed in two cases. In Case 1, after the specimens were isotropically consolidated, depressurization was conducted at a rate of 0.5MPa/min. Then, repressurization was performed at the same speed after finishing MH dissociation. These pore water pressure histories correspond to real production of MH in recovering pore water pressure after production. In Case 2, after specimens were isotropically consolidated, initial shear stress was applied at an amount greater than the host sand but less than MH bearing sand. Depressurization is then conducted in the same way as Case 1 and at the same speed. After finishing MH dissociation, pore water pressure was increased at the same rate. This test was done to simulate the element in the vicinity of the production well, where the stress condition is close to failure. In Figure 10, the specimen failed during repressurization when the stress path reached the failure line of the host sand. In Figure 11, the relationship between the temperature and pore water pressure during the depressurization process is shown. As can be seen, temperature decreased suddenly when the pore pressure was decreased to the value of the state boundary curve. This is due to the temperature absorption phenomena of MH during dissociation. Also, at a pressure of 3MPa dissociation and reformation repeats, and when dissociation is complete the temperature of the specimen rises to room temperature. Figure 12 shows the relationship between the effective stress ratio and active strain during dissociation tests of MH for Case 2. Point (a) corresponds to the point before dissociation, where initial shear stress has been applied. Point (b) corresponds to the point where pore water pressure was decreased from 10MPa to 3MPa.

1014



4

Point (c) corresponds to the point where MH is dissociated. Point (d) corresponds to the point where the specimen failed due to an increase in pore water pressure (repressurization). Photo 1 shows the specimen at each axial strain during Figure 12. In Case 2, the specimen failed during repressurization when the stress path reached the strength of the host sand. From the photo, it can be seen that from point (d) a shear band appeared in the specimen and failure occurred. Figures 13 and 14 show the volumetric strain and maximum shear stress contour obtained during Photo 1. From point (a) to (c) the specimen was consolidated and the volume was compressed. At (d), volumetric dilation occurred in the shear band and local deformation was observed clearly.

5 CONCLUSIONS

(1) In drained shear tests on host sands, initial stiffness and strength decreased with increasing fines content and there was a strong trend for contraction of volumetric strain. (2) For all specimens, the initial stiffness and peak strength of those containing MH increased due to MH`s cementation force, and volumetric strain behaved in a more dilative manner. However, both initial stiffness and strength decreased with increasing fines content of host sands. (3) In plane strain shear tests, Tc and Toyoura sand as host sands with and without MH were compared. Due to the existence of MH, initial stiffness and strength increased in both materials, however the tendency was more apparent in Toyoura sand compared with Tc. (4) The local deformations occurred more clearly in Toyoura sand, compared with fine material. It also appeared more clearly when the specimen contained MH. (5) During depressurization, marked deformation was not observed, because of an increase of effective stress. However, after depressurization, repressurization caused the specimen to fail in the case of high initial shear stress conditions.

6 ACKNOWLEDGEMENTS The first half of the present work was done as the activity of Research Consortium for Methane Hydrate Resources inJapan (MH21 Research Consortium) by the Ministry of Economy and Industry and the latter half was supported by KAKENHI 20246080 by the Ministry of Education and Science in Japan. The authors would like to express their sincere thanks to their supports.

7 REFERENCES

Hyodo, M., Nakata, Y., Yoshimoto, N. and Yoneda, J. 2008. Shear strength of methane hydrate bearing sand and its deformation during dissociation of methane hydrate. Proc. of 4th Int. Symp. on Deformation Characteristics of Geomaterials, 549-556.

MH21 Research Consortium 2012. http://www.mh21japan.gr.jp/Yamamoto, K. 2009. Production Techniques for Methane Hydrate

resources and Field Test Programs. Journal of Geography, Vol.118, No. 5, 913-934.

Yoneda, J., Nakata, Y. 2011. Deformation of deep seabed during dissociation of methane hydrate. Proc. The 14th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering. ISSMGE. Paper ID290.

Figure 12 Relationship between stress ratioand axial strain (Case2)

0.0

0.5

1.0

0 5 10 15

Str

ess

ra

tio

(

' 1 -

' 3)/

( ' 1

+

' 3)

Axial strain a(%)

Case2(a)

(c)

(d)

(b)

SMH=0.0%

1 2 3 4 5 6 7 80

2

4

6

8

10

12

Temperature T　(oC)

Stability boundary

After depressurization

(Water-Methane)

(Methane hydrate) Initial condition

Case1Case2

Por

e p

ress

ure

u (

MP

a)

Figure 11 Temperature and pore pressurepath in depressurization test

(a)a=3.5% (b)a=4.0% (c)a=7.2% (d)a=12.7%

Figure 13 Contour of volumetric strain at each axial strain (Case2)

Figure 14 Contour of maximum shear strainat each axial strain (Case2)

0255075100

(%)

max

Photo 1 The specimen image at each axial strain(Case2)

-25025

(%)

v

(a)a=3.5% (b)a=4.0% (c)a=7.2% (d)a=12.7%

(a)a=3.5% (b)a=4.0% (c)a=7.2% (d)a=12.7%

Figure 10 Stress path in repressurization test

Figure 9 Stress path in depressurization test

0

1

2

3

4

5

6

0 5 10 15 20Prin

cipa

l stre

ss d

iffer

ence

(

' 1 -

' 3)/2

(MP

a)Mean principal stress ('1 + '3)/2 (MPa)

Case1Case2

Failure envelope of MH bearing sand

Failure envelope of Host sand

Depressurization

0

1

2

3

4

5

6

0 5 10 15 20Prin

cipa

l str

ess

diffe

renc

e (

' 1 -

' 3)/2

(MP

a)

Mean principal stress ('1 + '3)/2 (MPa)

Water pressure recovery

Failure envelope of Host sand

Failure

Case1Case2

1015

A Simplified Contact Model for Sandy Grains Cemented with Methane Hydrate

Un modèle simplifié pour les contacts entre grains de sable cimentés par hydrates de méthane

Jiang M., Liu F., Zhu F., Xiao Y. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering (Tongji University), Ministry of Education, Shanghai 200092, China

ABSTRACT: Methane hydrates (MHs), regarded as one of the most promising future energies, extensively occupy the voids of soil deposits in permafrost regions and deep seabed. The presence of MH largely changes the macro-mechanical properties of the host deposits due to MH bonds among soils particles. This study introduces a new contact model for soil grains cemented by MH. This model was experimentally calibrated from two rods cemented by different materials. The bond failure criterion was then related to the strength properties of MH considering the effects of temperature, confining pressure, density and saturation of MH. The new model was implemented into a discrete element code to simulate the mechanical response of a MH bearing soil specimen subjected to biaxial loading conditions. The comparison between the simulation and experimental results shows that the new contact model can qualitatively capture the effects of MH cementation. Strain softening and shear dilation of soils cemented with MH become remarkable due to the presence of MH bonds. The cohesion of the MH bearing deposit substantially increases with the hydrate saturation, while the internal friction angle is less affected.

RÉSUMÉ : Les hydrates de méthane (MH) sont considérés comme une source énergétique potentielle fondamentale pour le futur. Ils abondent dans les régions avec permafrost et sur les fonds marins profondes. Les MH influencent grandement les propretés macro-mécaniques des sols qui les contiennent en raison de liens qu’ils forment entre les particules des sols granulaires. Dans ce papier, on introduit un nouveau modèle pour les liens entre particules de sol cémentes par MH. Le modèle a été calibré utilisant des données expérimentales de pairs de barres métalliques (barres de Schneebeli) cimentées par diffèrent matériaux. Le critère de rupture pour les liens (bonds) a été corrélé aux paramètres de résistance au cisaillement des MH incluant les effets de la température, la pression de confinement, la densité et la saturation des MH. Le modèle a été implémenté dans un logiciel aux éléments discrets pour simuler laréponse mécanique des échantillons de sols contenant des MH chargés en conditions biaxiales. La comparaison entre les résultatsexpérimentaux et numériques montre que le nouveau modèle reproduit qualitativement les effets de la cimentation des liens MH. On a observé que les liens MH causent un adoucissement et dilatation remarquables. La cohésion et l’angle de frottement des sols contenant MH augmentent avec le dégrée de saturation des hydrates avec une augmentation plus significative pour la cohésion que pour l’angle de frottement.

KEYWORDS: Methane hydrate bearing sands; bond; contact model; distinct element method; granular material

1 INTRODUCTION

As promissing resource of future energy, methane hydrates (MHs) are extensively found in voids of sediments situating in seabeds and permafrost regions at low temperatures and high pressures (e.g., Kvenvolden, 1988). They greatly enhances the strength of the host sediments. Moreover, MHs are prone to dissociation due to change in envrionmental conditions and human activities (e.g., installation offshore infrasturcutes). Serious geohazards such as marine landslides are likely triggered by instability of methane hydrate bearing sediments (MHBS). Unfortunately, the mechanism of these geo-hazards is poorly understood due to the lack of robust constitutive model of MHBS in addition to limited experimental data.

A variety of influencing factors on mechanical properties of MHBS were investigated using special triaxial compression aparatus (Hyodo et al. 2005; Masui et al. 2005). In particular, the hydrate habit (i.e., the distribution of MHs in the pore scale) strongly affects properties of MHBS. For example, hydrates acting as inter-particle cementation cause larger strength and stiffness than pore-filling hydrates. It implies that hydrate habit should be featured in a robust constitutive model. However most models derived from laboratory tests at the macro scale are unable to build the connection between macroscopic properties of MHBS and micro structure of hydrate at the pore scale.

In the contrast, the distinct element model (DEM) proposed by Cundall and Strack (1979) provides a solution for simulating

hydrate habits at the grain scale. Waite et al., (2009) identified three habits: (1) pore filling with hydrates nucleating in the pore without bridging two or more soil grains together; (2) load bearing with hydrate bridging nearby soil grains and taking part in the strong force chains of the granular assembly; and (3) cementation with hydrates cementing at inter-particle contacts as illustrated in Fig. 1(a). The first type naturally turns into the second when the hydrate saturation exceeds 25-40%. Pore-filling hydrate has been successfully modeled by Brugada et al. (2010) and Jung et al. (2012) using DEM. However the model of cementation-type hydrate remains unsolved partially due to the difficulty in quantifying the strength of MH bonds. This constitutes the strong motivation of this paper. The objective of this study is to develop a suitable bond contact model for MHBS, which is of critical importance to produce meaningful macro-mechanical response of MHBS via DEM.

2 A BOND CONTACT MODEL FOR MHBS

2.1 A conceptual bond contact model and laboratory data

Fig. 1 shows a conceptual bond contact model (Jiang et al. 2006) of two disks with radii R1 and R2 bonded at a finite width B and a central thickness t. A dimensionless parameter is defined in terms of common radius R =2R1R2/(R1+R2):

RB (1)

1016


Figs. 1(b) to (d) illustrate how this model responds to the normal contact force Fn, the shear contact force Fs, and the contact moment M. These contact forces can be computed as:

],min[ nbnnn RuKF (2a) ] (2b) ,min[ sbsss RuKF ],min[ rbm RKM (2c)

where min[•] is the operator taking the minimum value; un, us and are the overlap, relative shear displacement, and relative rotation angle; Kn, Ks and Km= Kn R /12 are the normal, tangential and rolling contact stiffness; Rnb, Rsb, and Rrb are the normal, shear and rolling bond strength.

For simplicity, the inter-particle rolling resistance is ignored at contacts with broken bonds or without bonds. At these contacts, the linear contact law applies:

nnn uKF (3a)

],min[ nsss FuKF (3b) where is the inter-particle friction coefficient; K’n and K’s are the post-failure normal and tangential contact stiffness.

soil grain

tension

(d)(c)

(b)(a)

M

us

Fs

Fn

un

Rrb

Residual strength

soil grain hydrate

BRnb

1Kn

Kn1

Rsb

1Ks Residual

moment1Kr

compression

R1R2

t

Figure 1. Schematic illustration of (a) MH bonded soil grains and its response: (b) Fs against un; (c) Fs against us; and (d) M against

Jiang et al. (2012a, 2012b) performed microscopic contact mechanical test to calibrate the model parameters for a large number of aluminous rod pairs (with a diameter of 12 mm and a length of 50 mm) bonded by epoxy and cement. Their study resulted in a generic criterion for bond failure, forming a strength envelope in a three-dimensional space with axes being Fn, Fs, and M. The projection of the envelope in Fs-M plane can be approximated as an ellipse as the following:

12

2

2

2s

rbsb RM

RF (4)

The ellipse varies in size with increasing Fn. That is, Rsb and

Rrb are functions of Fn. For the case of thick bonds, they can be computed as follows based on experimental findings if only normal forces are applied on the bond:

n

tnnctnsssb RFFRRFLfR )]/()[()( m

(5a)

tnnctnrrrb RFFRRFLfR )]/()[()( (5b) where Rt and Rc are the bond tension and compression strength, respectively, which can be obtained from tension/compression test of the cementation materials. Ls and Lr are the slopes of the straight line linking Rt to the peak shear strength or peak rolling resistance on the projection plane. fs, fr, n and m are fitting parameters to the experimental data. They can be calibrated from contact mechanical tests as Jiang et al. (2012a, 2012b) did recently. Fig. 2 illustrates their test results with a comparison to the prediction by Eq. (5).

Since MH remains stable in very extreme conditions, it is still a challenge to directly measure the strength parameters for hydrate bonds. Thus, in addition to test data from similar bond materials, assumptions are necessary for indirectly determining model parameters for MH bonds. This will be explained later.

-5 0 5 10 15 20 250

3

6

9

12

15 Eq. 5a

Test data (Jiang et al. 2012a,b) Cement-bonded Epoxy-bonded

R rb (k

N·m

)

Fn (kN)

(a)

-5 0 5 10 15 20 250

2

4

6

8(b) Eq. 5b

R rb (

kN·m

)

Fn (kN)

Test data (Jiang et al. 2012 a,b)Cement-bonded Epoxy-bonded

Figure 2. Bond strength envelopes derived from laboratory data: (a) strength envelope for Rsb; and (b) strength envelope for Rrb

2.2 Determing from hydrate saturation

The parameter relates to a given value of hydrate saturation SH, which is defined in a two-dimensional context as the ratio of the area of voids occupied by MH, AH, to the total void area, AV. The area of voids occupied by the ith MH bond is:

2

2 2 1 2arcsin(4bi iA R )

2

(6)

where we assume (1) the radii of the two bonded particles equal to

iR (i.e. neglecting the different curvatures of the particles); and (2) the bond thickness is negligibly small. The total area occupied by hydrate bonds, Ab, can be found by summation over all the bonds. Saturation attributed to pore-filling and bonding hydrates are denoted as SHb and SHp, respectively. SHp, generally equal to 20-30% (Masui et al. 2005), can be regarded as the threshold value of hydrate saturation at which MHs start to bond sand grains. Accordingly,

1

(1 )(1 )

mpbH

H Hb Hp p Hp bi HpiV

eAAS S S e S A SA A A

(7)

where m is the total number of bonds; A is the total area of a cross section of the sample; ep is the planar void ratio. Eq. (7) provides a non-linear relationship which depends on the state of compaction of the sample (e.g. relative density) which rules the particle packing and therefore the value of m. Fig. 3(a) shows a sample curve achieved for the case of a dense soil sample (e.g., ep= 0.21) consisting particles with diameters ranging from 6 to 9 mm forming a gradation curve as shown in Fig. 3(b).

0 10 20 300.0

0.3

0.6

0.9

1.2

Para

met

er β

nding saturation SBo4 6 8 10

0

20

40

60

80

100

12

Perc

enta

ge in

mas

s sm

alle

r tha

n (%

)

Particle diameter (mm)

(b)

ep =0.21

(a)

H-SH0 (%)

Figure 3. (a) Model parameter at different hydrate saturation for a dense sample (ep=0.21) with (b) a given gradation curve

2.3 Contact stiffness

Soil grains with the Young’s modulus ranging from 50 to 70 GPa can be regarded as rigid particles relative to MH. Thus, for MH-bonded grains, Kn = BE/t. The Young’s modulus of MH, E, depends on temperature T, confining pressure pc and density according to the test data (Hyodo et al. 2005). The regression analysis of these data yields the empirical formula:

03 +4950.50 1.98 1821.78a c a wE p p p T T (8)

1017


where pa is standard atmospheric pressure (i.e., 0.1 MPa); w is the water density at 4C; T0 is the reference temperture (1C is used herein).

The tangential contact stiffness Ks then relates to Kn based on PFC2D Guidelines (Itasca, 2004) by Ks = 2/3 Kn.

0.72 0.76 0.80 0.841.8

2.1

2.4

2.7

1.5MPa

4MPa

Elas

tic m

odul

us (1

08 Pa)

Density g/cm3

pc =8MPa

T= -300C

(a)

0.72 0.76 0.80 0.842.2

2.3

2.4

2.5

2.6

2.7

50C-100C

Elas

tic m

odul

us(1

08 Pa)

Density(g/cm3)

p0 = 8MPa

T=-300C

(b)

Figure 4. Elastic modulus of pure MH measured (a) at a given temperature but varied confining pressure; and (b) at a given confining pressure but varied temperatures (data from Hyodo et al. 2005)

2.4 Bond tension/compression strength

The bond tension and compression strength (i.e., Rt and Rc in Eqs. (4) and (5)) can be computed from the tension/compression strength, qmax,t and qmax,c, of pure MH specimen subjected to a given confining pressure pc, i.e.,

max,tR Rq t (9a) max,cR Rq c (9b)

As partially shown in Fig. 5, the peak deviator stress qmax,c

obtained from the compression triaxial test on pure MH specimens is a function of T, pc and (Hyodo et al. 2005). The regression analysis of these data yields another empirical formula:

max, 00.81 2.08 184.16 134.65c a c a wq p p p T T (10a)

Eq. (9) is assumed to hold for tension triaxial test as well, leading to the following:

max, 00.81 2.08 184.16 134.65t a wq p p T T (10b) where

00.55 1.15 100.09 74.39c a wp p p T T (11)

2.5 Fitting parameters

As already demonstrated in Fig. 2, the shape of the bond strength envelope is controlled by the cementation materials, resulting in different values of fitting parameters used in Eqs. (4) and (5). Direct calibration of those parameters are rather difficult for hydrate bonds. We assume that bond strengths are dominated by the strength properties of cementation materials. Fig. 6(a) shows the yielding curves of different materials. The yielding curve is left skewed for cement-based materials and right skewed for epoxy. Unfortunately, we are not able to present the yielding curve for MH due to insufficiency of test data available. As an ice-like material, MH was found similar to ice in some physical properties (Solan et al. 1998; Dvorkin et al. 2000) and mechanical properties (Nabeshima et al. 2003; Choi et al. 2009). Instead of MH, the yielding curve of ice is plotted in Fig. 6(b) for comparison, which displays left skewed. We concluded that the MH bonds produce a bond strength envelope similar to cement bonds. The fitting parameters calibrated using cement-bonded grains are used for MH-bonded grains. Eq. (5) is re-written as:

0.381.16 0.498 ( ) [( ) / ( )]sb n t c n n tR F R R F F 0.38

RR

(12a) 1.13 0.96 ( ) [( ) / ( )]rb n t c n n tR F R R F F (12b)

Suppose a MHBS sample collected at 800 m below the sea level (giving a pore water pressure of 8 MPa applied on MH bonds in the sample) at 5C. The gradation curve of the soil grains is assumed to be Fig. 3(b) and the initial void ratio of the sample is 0.21. For a given SHb, the parameter is determined from the chart presented in Fig. 3(a). Rt and Rc computed from Eqs. (9) to (11) are then substituted into Eq. (12) to obtain the bond strength envelope. Fig. 7 illustrates a series of strength envelopes at different levels of hydrate saturation. As expected, the envelope expands omothetically with SHb.

0 2 4 6 8 100

5

10

15

20

25

qm

ax (M

Pa)

Comfining pressure p0 (MPa)

T = -30 ℃ = 0.8g/cm3

(a)

10 0 -10 -20 -30 -40

0

5

10

15

20

25

High purity

Low purity

q max

(MPa

)

Temperature (℃ )

p0 = 8MPa

= 0.8g/cm3

(b)

Figure 5. The maximum deviator stress of pure MH measured (a) at a given temperature but varied confining pressure; and (b) at a given confining pressure but varied temperatures (data from Hyodo et al. 2005)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

2.5

3.0

Cement-basedEpoxy resin

Hussein & Marzouk 2000Kupfer et al., 1967Ellyin et al., 2005

3/3max

(1-

3)/3m

ax(a)

-0.3 0.0 0.3 0.6 0.9 1.20.00

0.05

0.10

0.15 Nadreau & Michel 1986

3/

3max

(1-

3)/3m

ax ice

(b)

Figure 6. Yielding curves obtained from experiments for: (a) cement-based materials and epoxy resin; and (2) ice

-20 -10 0 10 20 30 400

5

10

15

20

25

15%24%

SHb=41%

R sb (

kN)

Normal force Fn (kN)

-20 -10 0 10 20 30 400

10

20

30

40

15%

24%SHb=41%

R rb (N

·m)

Normal force F

(a) (b)

n (kN)

Figure 7. Bond strength envelopes at different hydrate saturation: (a) relationship between Rsb and Fn; and (b) relationship between Rrb and Fn

3 DEM SIMULATION OF BIAXIAL TESTS ON MHBS

The proposed model for MHBS was implemented into the commercial code PFC2D for simulating the biaxial compression test on a MHBS specimen. For inter-particle contacts with broken bonds or without bonds, = 0.5, K’n = 3.0×108 N/m, K’s = 2.0×108 N/m. The model parameters for other contacts with intact MH bonds were determined as explained in Section 2.

A 40×80 cm virtual specimen was first generated using the multi-layer with under-compaction method (Jiang et al. 2003). The resulted specimen with an initial planar void ratio of 0.21 was composed of 6000 disks with radii ranging from 6 to 9 mm, forming a gradation curve shown in Fig. 3(b). The specimen was consolidated by applying an isotropic pressure of 1 MPa until force balance was maintained. The gravity was ignored in the whole simulation. Bonds were then assigned to the contacts of particles. As the force system was balanced, the top wall moved downward at a speed of 5% per minute to simulate the displacement-controlled shearing process. Frictionless rigid walls were used in the simulation in stead of flexible walls to reduce the computational time. The boundary type selected will not affect the stress-strain response, while it however will change the formation of shear bands.

1018


4 SIMULATION RESULTS

The simulation was conducted on four specimens at SHb = 0%, 15%, 24% and 41% at a pore water pressure of 8 MPa and a temperature of 5C in order to compare to the experimental data published in Masui et al. (2005). SHp is assumed to be approximately 26% according to the data in Masui et al. (2005).

Fig. 8 shows the simulated stress-strain response at an effective confining pressure of 1 MPa and test results obtained under the same conditions. Although the simulation can not quantitatively reproduce the tests, it captures the essential features such as strain softening at SH > 26%. At higher SH, the peak strength is mobilized when the axial strain exceeds approximately 3%, and the residual strength coincides at a large strain regardless of hydrate saturation due to complete breakage of hydrate bonds. This agrees well with experimental data. However the peak deviator stress obtained from the simulation is lower than the test results. Besides the difference between biaxial and triaxial tests, one of the reasons is that the bond tension and compression strength could be underestimated in the model. The size of the specimen used in material strength tests is much larger than inter-particle bonds in MHBS. The strength measured from a large specimen is much lower than that of a much smaller specimen.

Fig. 9(a) presents an example of the stress-strain behavior under different confining pressures, which leads to a relationship between the peak strength parameters and SH as depicted in Fig. 9(b). The presence of hydrate cause the increase in cohesion, while no significant change in the internal friction angle is found associated with increasing SH. This agrees well with the experimental observation (Masui et al. 2005). However the friction angle obtained from the simulation (approximately 20) is lower than the test data (approximately 30). This could be improved by introducing the inter-particle rolling resistance in the model. The micro parameters associated with the rolling resistance can be first calibrated from a simulation on a specimen without MH bonds in order to reach high friction angle. These parameters set are then brought into MHBS model. Considering the inter-particle rolling resistance will result in a higher peak deviator stress, which better matches the test data as shown in Fig. 8(b).

0 4 8 12 160

1

2

3

S=67%

0-26%

40%50%

Dev

iato

r Stre

ss (

MPa

)

Axial Strain (%)

(a)

0 4 8 12 160

3

6

9

26.4%

40.9%

50.1%

Dev

iato

r stre

ss (M

Pa)

Axial strain (%)

SH=67.8%(b)

Figure 8. Deviator stress vs. axial strain: (a) DEM simulation; and (b) triaxial test results performed by Masui et al. (2005)

0 2 4 6 8 10 10.0

0.5

1.0

1.5

2.0

2.5

2

SH=50%

0.7MPa

0.52MPa

1.1MPa

Dev

iato

r stre

ss (M

Pa)

Axial strain (%)

1.5MPa(a)

0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

16

18

20

22

24

Cohesion

Fric

tion

angl

e (o )

Coh

esio

n (M

Pa)

SH (%)

Friction angle

(b)

Figure 9. Simulation result (a) deviator stress vs. axial strain at different confining pressure for a specimen with SH=50%; and (b) peak strength parameters at different SH.

5 CONCLUSIONS

This paper proposed a two-dimensional bond contact model of MHBS for considering the bonding effect of MH. The bond strength envelope was partially derived from laboratory data. The model parameters are related to the hydrate saturation,

confining pressure, temperature and density of MH. Using this model, the DEM simulation of the biaxial test is capable of capturing the major mechanical response of MHBS specimen such as strain softening and shear dilation at high hydrate saturation. This study can help to better understand the connection of the microscopic formation habit of MH to macroscopic mechanical behaviors of MHBS.

Though the DEM simulation produced results qualitatively comparable to available test data, quantitative agreement remains still a challenge. The current model ignores the bond thickness, which however affects the hydrate saturation and the bond strength parameters. Consideration of inter-particle rolling resistance in the model will improve the model performance. Moreover, the size effect on the bond strength remains unclear and deserves more caution. Further investigation on these issues is definitely needed in the future work.

ACKNOWLEDGEMENTS

This work is funded by China National Funds for Distinguished Young Scientists (No. 51025932), and the EU FP7 IRSES grant (No. 294976).

REFERENCES

Brugada J. et al. 2010. Discrete element modelling of geomechanical behaviour of methane hydrate soils with pore-filling hydrate distribution. Granular Matter 12(5, SI): 517-525.

Cundall P.A. and Strack O.D.L. 1979. A discrete numerical model for granular assemblies. Géotechnique 29(1), 47–65.

Ellyin F. and Xia Z.H. 2006. Nonlinear viscoelastic constitutive model for thermoset polymers. Journal of Engineering Materials and Technology 128:579-585.

Hussein A. and Marzouk H. 2000. Behavior of high-strength concrete under biaxial stresses. ACI Materials Journal 97(1):27-36.

Hyodo M. et al. 2005. Basic research on the mechanical bahavior of methane hydrate-sediments mixture. Japanese Geotechnical Society 45(1):75-85.

Jiang M.J. et al. 2003. An efficient technique for generating homogeneous specimens for DEM studies. Computers and Geotechnics 30(7): 579-597.

Jiang M.J. et al. 2012a. Contact behavior of idealized granules bonded in two different interparticle distances: An experimental investigation. Mechanics of Materials 55: 1-15.

Jiang M.J. et al. 2012b. An experimental investigation on the mechanical behavior between cemented granule, Geotechnical Testing Journal (ASTM) 35(5): 678-690.

Jiang M.J. et al. 2006. Bond rolling resistance and its effect on yielding of bonded granulates by DEM analyses. International Journal for Numerical and Analytical Methods in Geomechanics 30(8): 723-761.

Jung J. et al. 2012. Stress-strain response of hydrate-bearing sands: Numerical study using discrete element method simulations. Journal of Geophysical Research-Solid Earth, 117, B04202, doi:10.1029/2011JB009040.

Kupfer H. et al. 1969. Behavior of concrete under biaxial repeated loading. ACI Journal Proceedings 66(52): 656-666.

Kvenvolden K.A. 1988. Methane hydrate-a major reservoir of carbon in the shallow geosphere? Chemical Geology 71(1-3):41-51.

Masui A. et al. 2005. Effects of methane hydrate formation on shear strength of synthetic methane hydrate sediments. Proceedings of the 5th International Offshore and Polar Engineering Conference, Seoul, Korea: 364-369.

Nadreau J.P. and Michel B. 1986. Yield and failure envelope for ice under multiaxial compressive stresses. Cold Regions Science and Technology 13:75-82.

Waite W.F. et al. 2009. Physical properties of hydrate-bearing soils. Reviews of Geophysics 47, RG4003:1-38.

1019

Macro- and micro-FE modelling of wellbore damage due to drilling and coring processes

Modélisation par les éléments finis aux échelles micro et macro de l’endommagement dû au forage et au carrotage

Khoa H.D.V., Grande L., Jostad H.P. Norwegian Geotechnical Institute, Oslo, Norway

ABSTRACT: This paper presents the application of the finite element method to evaluate tensile fracturing at different scales in theformation during drilling and coring operations. The first part focuses on evaluating the formation damage process at the macro scale.Both full 3D and axisymmetric macro FE-model are established in order to identify the potential damage mechanisms, to study thesize of damage zone as well as to indentify the critical stress changes causing failure cracks at one specific location close to the wellbore tip. In the second part of the paper, the potential mechanisms of damage are investigated in detail at the micro scale (i.e. grain scale) by using a complex 2D micro FE-model reproducing a realistic grain structure taken from the Scanning ElectronMicroscope (SEM). The stress changes calculated from the macro FE-model are applied at the boundaries of the micro FE-model to simulate the effects from drilling and coring operations. The calculated results show that the 2D micro FE-model is closer to explain the formation damage observed around the wellbore and more pertinent to investigate the physical processes of damage, while it isalmost impossible with the macro FE-model.

RÉSUMÉ : Cet article présente l'application de la méthode des éléments finis pour évaluer une rupture en traction à différenteséchelles dans une formation rocheuse au cours des opérations de forage et de carottage. La première partie se concentre surl'évaluation du processus de détérioration de la formation à l'échelle macro. Les deux cas de modélisation 3D et de symétrie axiale par les éléments finis sont mis en place afin d'identifier les mécanismes d’endommagements potentiels et d'étudier la taille de la zone endommagée ainsi que pour identifier les changements de contraintes critiques causant une fissuration. Dans la deuxième partie del'étude, les mécanismes potentiels d’endommagement sont étudiés en détail à l’échelle micro (i.e. échelle de grain) en utilisant un modèle 2D complexe qui reproduit une structure de grain réaliste issue d’observations par microscope électronique à balayage (MEB). Les changements de contraintes calculés à partir de la modélisation macro sont appliqués aux frontières du modèle Eléments Finis à l’échelle micro pour simuler les effets de forage et de carottage. Les résultats calculés montrent que le modèle 2D micro est plus apte à expliquer la détérioration de la formation observée dans le puits de forage et plus pertinent pour étudier les processusphysiques d’endommagement qui sont presque impossibles à aborder avec le modèle macro.

KEYWORDS: Multiscale modelling, wellbore damage, finite element

1 INTRODUCTION

Drilling and coring operations disturb and generate stress changes in the rock surrounding the wellbore. These induced stresses, which are quite different in magnitude and sometimes orientation, as compared to the initial conditions, can cause a number of events such as wellbore stability and fracture initiation.

In general, well damage is governed by the in-situ stresses, pore pressure and rock strength. In addition to these dominant parameters, during the drilling and coring operations, wellbore stability may directly or indirectly be affected by the three following effects:

• Stresses from drill bit (shear stresses from torque, lateral stresses due to stress release due to drilling vibration and weight of bit); • Stresses released due to drilling: difference between mud pressure and the in-situ stresses and reduction by mud-fluid flowing into the formation; • Stresses released due to temperature reduction due to cooling of the formation rock near the wellbore by the colder mud-fluid.

Recent developments in geophysical logging i.e sonic scanner logging tool together with the Diapole Shear Radial Profiling algorithm have given new insight into the evaluation of 3D field of stresses and material properties around the wellbore (Sayers et. al, 2009). The information from such logs along the well are crucial with respect to predicting potential geomechanical challenges during drilling and coring operations. It was found that one possible mechanism of wellbore damage is tensile failure of the formation during effective stress unloading caused by radial stress release, pore pressure increase by mud-fluid flowing into the formation close to the wellbore tip as well as temperation reduction.

The paper focuses on applying the finite element method to evaluate tensile fracturing at different scales (multiscale) in the

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Figure 1. Field data used for study of formation damage. Depth of burial varies between two fields A and B, leading to pronounce difference in terms of quartz cementation. Also use of different muds leads to different mud cakes at wellbore walls.

Table 1. Material properties at wellbore at Field A and B. Parameters Field A Field B UnitDepth ~4100 ~2400 mTVD Porosity 16-25 27-32 %Young modulus 5 GPaPoisson’s ratio 0.15 0.27 - Cohesion 8.7 3.2 MPa Friction angle 33 29.1 degree Tensile strength 3.2 1.2 MPa

formation during drilling and coring operations. In the first part, an example of workflow will be outlined throughout the numerical modelling performed at two macro (continuum) and micro scales based on data from the two field cases: Field A and B. Figure 1 illustrates the locations of Field A and B where the Radial Diapole Profiling data is available. The material properties at the two field cases are given in Table 1. They basically differ from each other in term of burial depth, leading to different degrees of cementation between grains.

The second part will present the FE-results calculated by using the macro FE-model while the third part will discuss in detail about the application of the 2D micro FE-model for analysing and predicting tensile fracturing observed on a micro level (grain scale) in the formation during drilling and coring operations.

2 METHODOLOGY AND WORKFLOW

In the first stage a macro 3D FE-model was established for calculation of continuum stress changes around wellbore, which were mainly caused by lateral vibrations and torque from drill bit, radial stress release and mud-fluid flowing into the formation and temperature changes as illustrated in Figure 2.Several assumptions have been made:

• The rocks behave as a homogeneous material. Hence, its mechanical behaviour can be described by using a continuum mechanical approach; • The distances from the wellbore to the boundaries in the two horizontal (x) and (z) and vertical (y) directions were sufficiently large to avoid any effect of the outer boundaries on the stress changes close to the wellbore; • The elements were modelled assuming an elastic perfectly plastic, frictional-cohesive material that followed the Mohr-Coulomb failure criterion after the onset of yield. In addition to shear failure, the model also performs failure in tension. The material parameters used in the Mohr-Coulomb model are summarised in Table 1.This FE-model was applied for the detailed study of the two

Field cases A and B presented in Figure 1. A transient pore pressure analysis was performed in order to study the time dependent dissipation of excess pore pressure due to the mud-fluid flowing into the formation. In general, a total of five numerical simulation steps need to be performed in order to

evaluate the stress changes in the rock surrounding the wellbore during drilling and coring operations:

a. Initial stresses are first generated by applying the in-situ stresses to the FE-model without the wellbore. The considered stress states correspond to 2400 m “true vertical depth” (TVD) and 4100 m TVD below mean sea level for Field B and Field A, respectively; b. The wellbore is then excavated under the hydraulic support from the net mud pressure to simulate the drilling stage; c. In this step, forces at the drill bit are applied. All forces from cutters, bit body and gauge pads are summed and applied as resulting forces at the centre of the drill bit; d. Then the mud-fluid infiltrates into the formation and gives an applied increase of pore pressure around the wellbore;e. Finally, the reduction of the temperature is simulated by reducing the volumetric strain.

From a parametric study the mud temperature was found to be important with respect to generation of tensile fractures which are the most plausible failure mechanism. Downhole temperature logs measured at the drill bit were available, but these did not cover the entire history of temperatures during the various stages of drilling and circulations in the period between drilling and logging of the actual intervals. Due to the large impact of temperature on failure and the poorly documented temperature history of the study intervals, an assumed constant temperature change is applied one wellbore radius into the formation in order to simulate the cooling of formation during process of drilling. The stress situation one radius into the formation was evaluated with respect to fracturing due to tensile failure.

WOB

Fbit

From drill bit

Drill bit

Mud cake

TOB

From temperature

T within 1 radius from wellbore side

From mud

v

Borehole(r ×h : 0.108 × 1.5 m

infiltration of mud-fluid

y

A

y

zx

A

Figure 2. Full 3D FE modelling of different loads due to drill bit torque and axial load, mud-flow into formation and temperature change within one radius from wellbore wall.

1021


Vertical stress, V

Hor

izon

atls

tress

, h

Hor

izon

atls

tress

, h

Pore pressure, u

Rotation fixities

Beam elements

Figure 3. Cathodoluminescence SEM picture (top) from Storvoll (2004) and equivalent 2D micro FE-model (bottom) used for studying formation damage around wellbore during drilling and coring operations.

Then in the second stage the stress changes obtained from the macro FE-model are applied at the boundary of a micro FE-model in order to investigate tensile fracturing on the micro level. Figure 3 shows the FE-model established based on a high resolution cathololuminescence SEM picture of well cemented sandstone (from Storvoll, 2004), where authigenic quartz cement can clearly be separated from the original grains. For the deeply buried Field A which has undergone both mechanical and chemical compaction the cemented areas were activated at a stress condition corresponding to a burial depth of about 4.1 km corresponding to depth for onset of quarts cementation at temperatures of 70 - 800C (Bjørlykke, 1989). Field B has undergone purely mechanical compaction due to the shallow burial depth (< 2.4 km), and no cement was applied for this model.

-10

0

10

20

30

40

50

60

0 1 2 3 4 5 6

Effe

ctiv

e st

ress

es (M

Pa)

Loading step

Sig'_v

Sig'_hSig'_v Coring

Sig'_h Coring

Sig'_v DrillingSig'_t Drilling

Approximate stress at 2500 mbefore quarts cementationAssuming hydrostatic pressure

Max possible stress at 4100 m TWDAssuming hydrostatic porepressure

Final ef fective stress af ter coring and retrieval

Measured in-situ stress at 4100 m TVD

Uncertain stresspathPore pressure history is not known

Final ef fective stress in formation af ter drilling and temperature reduction

Coring and

drilling

Mechanical and

chemical compaction

Mechanical

compaction

Figure 4. Idealized stress paths for Field A. Stress changes during drilling and coring process are based on “macro” stresses released at material point A plotted in Figure 5.

Table 2. Simplified burial history applied in micro FE-model. Step Description V

(MPa) h(MPa)

u(MPa)

1 At 2500 m, no cement 55 40 252 At 4100 m, with cement 90 77 413 At 4100 m, in-situ state 90 85.8 75.8 4 Mud pressure & mud-flow 88.4 82.8 82.2 5 Temperature reduction (0C)

- 2.4 87.9 82.4 82.2 - 9.5 86.4 81.4 82.2 - 19.1 84.4 80.0 82.2 - 33.3 82.2 81.4 82.2 - 57.1 76.3 74.4 82.2 - 66.7 74.4 72.7 82.2 - 71.4 73.5 72.6 82.2 V, h are total vertical and horizontal stresses, respectively u is pore pressure

Figure 4 shows an idealized loading path being applied to the micro FE-model at Field A. The stress changes during drilling and coring processes are based on the “macro” stresses released at the material Point A located at a distance of one radius into the formation from the wellbore wall and two radius up from the wellbore tip (see Figure 5). In the process of coring there is first a total vertical stress unloading. This corresponds toremoving the weight of overburden during coring. Finally, the horizontal stress is completely unloaded. This corresponds to a situation after coring and core retrieval when pushing out the core from the core cylinder. In the formation adjacent to the wellbore the stress change is a combination of effect of pore pressure changes, temperature effects and stresses from the process of drilling. This is a complex process that requires FE modelling. Therefore the final load step has been directly taken from the calculated results of the macro FE-model. The effective vertical and effective radial stresses at Point A after drilling are shown in Figure 4. This includes unloading due to mudflow into the formation and cooling of the formation. Note that the final effective stresses shown in the figure corresponds to a cooling of about 700C, which is a higher temperature reduction than experienced in the field. Also, the tangential stress which is most critical with respect to micro fractures is not shown in this figure. Table 2 summarizes the burial history at Field A which is applied at the boundary of a micro FE-model in order to investigate tensile fracturing on the micro level.

3 RESULTS FROM MACRO FE-MODEL

From the results of the 3D modelling in it is found that at a distance of one radius from the wellbore wall the application of the drill bit forces gives minor effect in terms of stress changes compared to the effects from mud pressure, mud-fluid flowing into the formation and temperature reduction. In fact, damage only occurs very locally at the edges of the gauge pads and drilling cutters. Since the effects from the drill bit can be neglected, for a matter of modelling simplification and also to minimize the time necessary for processing, a macro axisymmetric FE-model has been established and used for studying the other effects with respect to the wellbore damage.

Figure 5 shows that the macro (continuum) analysis predictstensile fracturing within one radius into the formation (point A) after circa 670C of cooling. This value is not in agreement with field observations which indicate fracturing at much lower temperature differences as observed from the Radial Dipole Profiling data. Hence a further evaluation of tensile fracturing on a micro level was performed with a 2D micro FE-model (grain scale).

4 RESULTS FROM MICRO FE-MODEL

In both the Field A and the Field B case, it is found that onset and development of tensile fractures starts at a lower temperature reduction in the micro FE-model (Figure 6) then in the macro FE-model (see Figure 5). The micro FE-model thus seems to pick up formation damage that cannot be found when applying a macro FE-model, and this model is closer to explain the observed potential stiffness reduction based on the difference between the mud temperature and the formation temperature of 25-350C as indicated from downhole temperature log. There are however limitations in the 2D micro model and the results should only be used as a indications of

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that failure mechanisms may occur at earlier temperature stages than what is obtained by the classical macro (continuum) model.

9.5 0C

19.1 0C

57.1.6 0C

66.7 0C

33.3 0C 71.4 0C

Tensile failures

One physical explanation for this earlier development of tensile failure in the micro model compared to the macro model is partly linked to the stress history prior to coring or drilling. For cemented sandstone (i.e. Field A) the submerged weight of the overburden is initially carried by the original grain framework. When introducing cement to grain boundaries in the pore space during chemical compaction, the cement will initially be more or less stress free since the overburden pressure is already carried by the original grain framework. During unloading, since the stresses are smaller in the cement, it will therefore first reach tensile failure. Also, the micro model experience local stress concentrations which are not a part of the global continuum model. Temperature changes affect this local stress pattern differently compared to global unloading. This might explain the higher amount of tensile failures in the micro model also for the un-cemented Field B case. Further understanding of these physical processes is important to be able to explain and quantify core damage and formation damage in general.

Tensile failures

47.6 0C 52.4 0C 57.1 0C

59.5 0C 64.3 0C 66.7 0C

A A A

A A A

Figure 6. Development of tensile failures in formation predicted by micro FE-model.

5 CONCLUSIONS

The paper presents an approach for modelling the process of core and formation damages during drilling. Numerical simulations show that fracturing on a local micro scale seems to start at an earlier stage compared to macroscopic failure at the macro continuum scale. The main hypothesis or physical explanation for this earlier fracturing in the micro model compared to the macro model is partly linked to the stress and cementation history of sediments prior to coring or drilling, in interaction with mechanical, thermal and flow stresses induced during drilling, coring and production. This work is a promising contribution to a better understanding of physical processes resulting in core damage and formation damage in general. However, the work was limited to 2D micro modelling.

Figure 5. Development of tensile failures in formation predicted by macro FE-model.

Ideally, due to variability in possible grain structures and material properties and the 2D idealization of the 3D problem, the micro model should be extended to 3D and calibrated with laboratory data. There are also major uncertainties related to temperature history within the well and temperature distribution from the well into the formation. Further, available experimental data on the stress dependent anisotropy of sandstones shows very complex interrelationship between stress magnitudes and directions, and shear modulus. Future activities should also address better constitutive models for stress induced anisotropy in uncemented and cemented materials.

6 ACKNOWLEDGMENT

The authors would like to acknowledge Statoil and the Norwegian Research Council for contributing to and sponsoring this research and for providing data.

7 REFERENCES

Bjørlykke K.1989, Sedimentology and Petroleum Geology, Springer- Verlag Berlin Heidelberg.

Sayers C.M., Nagy Z., Adachi J., Singh V., Tagbor K. and Hooyman P. 2009. Determination of in-situ stress and rock strength using borehole acoustic data. Proceedings of the SEG International Exposition and Annual Meeting, Houston.

Storvoll V. and Bjorlykke K. 2004. Sonic velocity and grain contact properties in reservoir sandstones, Petroleum geoscience 10 (3), 215-226.

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1


Simulation tridimensionnelle par les éléments distincts du débit de décharge et d’écoulement gravitaire du matériau granulaire sableux

N. Kikkawa, K. Itoh & Y. ToyosawaNational Institute of Occupational Safety and Health, Japan

M. J. Pender & R. P. OrenseUniversity of Auckland, New Zealand

ABSTRACT: Reported herein are numerical simulations, using the discrete element method (DEM), of a bed of granular material having a moveable trapdoor over part of the underlying boundary. Kikumoto & Kishida 2003, and Kikumoto et al. 2003 measured the vertical stresses on a trapdoor and the adjacent boundaries in tests in which Toyoura sand was used to represent natural ground. The DEM simulation modelled well the vertical stress measured on the trapdoor when it was moved downward and also the vertical stresses on the boundaries adjacent to the trapdoor. Next the gravity flow of the sand was calculated when the trapdoor was suddenly removed; it was found that there was a complex dynamic response in the vertical stress on the boundary immediately adjacent to the opening created, but only modest changes further away. The motivation for these DEM simulations was the desire to understand better the processes involved in tunnel construction when there is a partial collapse of the face or the roof near the tunnel face. Although safety in the excavation of underground openings has improved markedly in recent decades, accident statistics for this type of work remain a challenge for the Japanese construction industry.

RÉSUMÉ: On présente ici des simulations numériques, à l'aide de la méthode d'éléments distincts (DEM), d'un lit de matériau granulaire ayant une trappe amovible sur une partie de la limite sous-jacente. Kikumoto & Kishida 2003, Kikumoto et al., 2003 ontmesuré les contraintes verticales sur une trappe et ses frontières adjacentes dans les essais où le sable de Tayoura a été utilisé pour représenter le terrain naturel. La simulation par DEM modélise bien la contrainte verticale mesurée sur la trappe quand il a été déplacé vers le bas et aussi les contraintes verticales sur les frontières adjacentes à la trappe. Ensuite, l'écoulement par gravité du sable a été modélisé lorsque la trappe a été soudainement retirée. Il a été constaté qu'il y avait une réponse dynamique complexe de la contrainte verticale sur la limite juste à côté de l'ouverture créée, mais seulement de légères modifications plus loin. La motivation pour ces simulations de DEM a été de mieux comprendre les processus impliqués dans la construction de tunnels lorsqu'il y a un effondrement partiel de la face ou sur les cotés près du front du tunnel. Bien que la sécurité dans l’excavation des cavités souterraines se soit nettement améliorée au cours des dernières décennies, les risques d’accidents dans ce type d’opération demeurent un défi pour l'industrie de la construction japonaise.

KEYWORDS: tunnel, trapdoor, gravity flow, discrete element method, sand

1 INTRODUCTION

At present, almost all mountain tunnels constructed in Japan are excavated utilizing the New Austrian Tunneling Method (NATM). This tunneling method was advocated by Prof. Rabcewicz from Austria in 1964 (Rabcewicz 1964a, b, 1965). In Japan, the method has been applied to tunnel construction since about 1978. On the other hand, tunnels in cities are often excavated utilizing a Shield Tunneling Method. Since adopting these methods there has been a substantial decrease in the number of accidents and fatalities in tunnel construction in Japan. However, there is still a relatively higher incidence of accidents during tunnel construction than in the construction industry in general. In tunnel construction, rock fall events in rocky ground and partial collapse from the roof and face in sandy ground are characteristic of the types of accident that occur.

Rock falls and the partial collapse at the face induce stress redistribution. This in turn leads to the formation of a ground arch above the failed roofs with accompanying local stress increases at the foot of the arch. These increased stresses may cause the collapse to extend to the whole tunnel itself. This mode of failure is more likely in sandy ground. Therefore, when

minor to modest collapse occurs it is very important to know how much the stress at the foot of the ground arch will increase above the values prior to the collapse.

The stress redistribution induced by lowering of a trapdoor has been tested and analysed (Terzaghi 1936, Murayama & Matsuoka 1971, Kikumoto & Kishida 2003, Costa et al 2009 etc.). Murayama & Matsuoka (1971) and Kikumoto & Kishida (2003) measured the earth pressure on the trapdoor and the surroundings during the lowering of the trapdoor. The Kikumoto & Kishida (2003) data are compared with the results from DEM modelling herein. Costa et al (2009) investigated the ground deformation mechanisms during lowering of a trapdoor with various ratios of the sand layer depth to the width of the trapdoor. They showed when the ratio is low, so modelling ashallow tunnel, the zone failure is limited to sand adjacent to and above the trapdoor. On the other hand, when the ratio is high, which represents a deep tunnel, the failure region is more widely spread beyond the width of trapdoor.

Earth pressure increases during a gravity flow were analysed using a two dimensional DEM by Kiyama & Fujimura 1983. They concluded that the earth pressure on the foot of the ground arch during the gravity flow is much higher than the static earth pressure. This finding is confirmed by the results of the DEM modeling presented in this paper.


Simulation tridimensionnelle par les éléments distincts du débit de décharge et d’écoulement gravitaire du matériau granulaire sableux

Kikkawa N., Itoh K., Toyosawa Y. National Institute of Occupational Safety and Health, Japan

Pender M.J., Orense R.P. University of Auckland, New Zealand

1024



2

In this study, we used a 3D particle flow code (PFC3D, Itasca 2012) to perform discrete element (DEM) simulations of the trapdoor experiments reported by Kikumoto & Kishida 2003, Kikumoto et al. 2003. In this work a bed of Toyoura sand 75~300 mm deep was placed by air pluviation into a box one metre square. The relative density of Toyoura sand achieved by this method was about Dr= 88%. Along the middle floor of the box, an instrumented strip 150 mm wide had been installed so that the vertical stress distribution could be measured. One part of this strip formed the trapdoor which could be moved downwards in a controlled manner or removed. With this arrangement the vertical stresses on the trapdoor and the floor of the container adjacent to the trapdoor could be monitored. These details are shown in the left hand side of Figure 1.

The first part of the DEM simulation calculated the vertical stress on the trapdoor and floor of the box as the trapdoor was moved downwards at a constant rate. In the second part of the simulation the vertical stresses on the floor of the container were calculated during gravity flow of the sand from the container after the sudden removal of the trapdoor.

2 OUTLINE OF DEM MODELLING OF THE TRAPDOOR AND GRAVITY FLOW TESTS

2.1 Three-dimensional Trapdoor and Gravity Flow Tests

The right hand side of Figure 1 shows the configuration of the DEM model. The dimensions of the soil container are almost the same as the experimental testing apparatus used by Kikumoto & Kishida 2003, except the depth of the sand layer is 150 mm whilst Kikumoto et al used various depths between 75 and 300 mm.

2.2 DEM analysis

Kikumoto & Kishida 2003 used Toyoura sand (D50= 0.20mm at Dr= 88%) in the experimental trapdoor tests. For the DEM analysis it is difficult to simulate the grain size distribution, particle shapes, and other properties of the sand. In this study, we used spherical particles for the DEM analysis, and calculated the normal and tangential stiffness coefficients, kn, ks, using P- and S-wave velocities that had been measured for Toyoura sand by Hori et al 2010. The stiffness coefficients are obtained from:

3

2psn VR

k

(1)

3

2sss VRk

(2)

where: Vp and Vs are respectively the P- and S-wave velocities of the granular medium , R is the average of sphere radius, s is the density of spherical particles. It is known that both normal and tangential interparticle stiffnesses may be functions of both wave velocities, but equations (1) and (2) assume that the normal stiffness depends only on Vp and the tangential stiffness only on Vs. Values for Vp and Vs of Toyoura sand of 403 and 254 m/sec respectively were obtained by Hori et al

2010 using the bender and extender element tests; thus kn and ks

were calculated to be 4.7 106 and 1.9 106 N/m, respectively. Table 1 shows the DEM parameters used in this study.

Table 1. DEM parameters used in this study Parameter Symbol Value Unit

Density of sphere s 2650 kg/m3

Mean radius of sphere R 10.5 mmRatio of maximum and minimum radius of sphere Rmax/Rmin 2.0 -

Normal stiffness kn 4.7 106 N/m

Tangential stiffness ks 1.9 106 N/m

Friction coefficient 0.5 -

Critical damping ratios ns 0.8 -

2.3 DEM procedure for trapdoor and gravity flow tests

The DEM procedures for the trapdoor and gravity flow tests are as follows:

Figure 1. Trap door and gravity flow testing apparatus (left), and DEM simulation (right).

Figure 2. Vertical stresses on the trapdoor and the floor next to the trapdoor against trapdoor displacement.

Figure 3. Experimental and DEM distributions of vertical stress at a trapdoor displacement of 2.0mm, with an overburden depth of 150 mm.

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

1

2

3

4

5

Ear

th p

ress

ure,

p (

kN/m

2)

D isplacement of trapdoor , d t (mm)

trapdoor (Kikumoto et al. 2003)

C (Kikumoto et al. 2003)

B (Kikumoto et al. 2003)

A (Kikumoto et al. 2003)

trapdoor (DEM)

C (DEM)

B (DEM)

A (DEM)

trapdoor

A B Cinitial earth pressure(=2.38kPa)

1000

unit: mm

1000

150 150 150 150 200200300150

550

500

500

trapdoor

A B C

-400 -300 -200 -100 0 100 200 300 4000

1

2

3

4

5

6E

arth

pre

ssur

e, p

(kN

/m2)

Hor izontal dis tance from the centre of trapdoor , h d (mm)

Kikumoto et al. 2003

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trapdoor

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1) Set-up the geometry of the walls and floor of the container and the elements on the floor where the contact stresses will be monitored and the details of the trapdoor as shown on the left hand side Figure 1. The trapdoor is 150 mm long and 150 mm wide and consists of three elements 50 mm long by 150 mm wide. The walls and floor of the container are assumed to be rigid. The normal and tangential stiffnesses between the floor elements and the spherical particles are the same as those of between the spherical elements. The friction coefficient between the spherical particles and the floor elements is 0.5, which is the same as that between the particles.

2) Generate the spherical elements above the floor in the upper part of the container to a depth of 150mm. Calculations are repeated under gravity loading until the displacement rates of the spheres approach zero and the bed has come to equilibrium. Knowing the maximum and minimum void ratios of Toyoura sand, the relative density of the sand bed was estimated to be 86%.

3) To simulate the trapdoor tests, the trapdoor was lowered at a constant displacement rate of 5mm/sec. The vertical stresses on the trapdoor and the floor next to the trapdoor were calculated.

For the gravity flow simulations, the trapdoor was removed instantaneously and dynamic vertical stresses on the floor next to the trapdoor were calculated.


First, the experimental data is compared with the DEM analysis data in the trapdoor tests and then the gravity flow simulation is discussed.

3.1 Trapdoor tests

Figure 2 shows the variation of vertical stress, p, on the trapdoor against the downward displacement, dt, and on the floor elements adjacent to the trapdoor. The depth of the sand layer was 150 mm, equal to the width of the trapdoor; the initial vertical pressure on the floor of the container was 2.38kN/m2.The plots compare the experimental data obtained by Kikumoto et al (2003) with the calculated DEM values. The centre of elements A, B and C are respectively 150mm, 300mm and 450mm from the centre of the trapdoor. It is seen that the vertical stress adjacent to the trapdoor is affected by the trapdoor movement but further away the influence of the movement is negligible. The vertical stress on the trapdoor rapidly decreased at the onset of the downward movement, and the experimental and DEM values corresponded well. The final

value when dt= 2.0mm is 0.62kN/m2 for the DEM results. On the other hand, the vertical stress on element A increased rapidly with the lowering of the trapdoor and then settles to a constant value of 3.0kN/m2 in the DEM. The initial rate of increase for the vertical stress for element A in the DEM results is different from that of the experimental data. This reason is not clear, but it may be related to the difference in dilation rate between Toyoura sand and spherical DEM elements.

The vertical stresses on floor elements B and C increased slightly and there is good agreement between the experimental data and the calculated DEM results.

Figure 3 shows the distribution of vertical stress laterally from the centre of the trapdoor when dt= 2.0mm. The vertical stress on the trapdoor decreased and increased on the floor next to the trapdoor. The DEM data show a sawtooth distribution not seen in the experimental data.

Figure 4 shows the surface settlement of the sand above the centre of the trapdoor when dt= 2.0mm at the depth of the sand layer was 150 mm. The experimental data was measured by the laser displacement scanner which accuracy was 1m (Kikumoto et al. 2003). It is, however, Figure 4 that reveals the most significant shortcoming of the DEM modelling as the experimental and calculated subsidence trough at the upper surface of the sand are very different; the surface settlement in

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Figure 4. Surface settlement of the sand above the centre of the trapdoor when dt= 2.0mm, layer thickness 150 mm. Figure 5. Vertical stress on the floor next to the trapdoor during

gravity flow.

Figure 6. Comparison of vertical stress distributions on the floornext to the trapdoor during trapdoor lowering and during gravity flow at the elapsed time of 0.3sec.

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the experimental test is much smaller than that from the DEM modelling. Again this could be caused by differences in dilatancy between the Toyoura sand and the spherical DEM elements. The particle shape of the Toyoura sand is well known to be sub-angular, so that in Toyoura sand it may be possible to establish the ground arch even in a loose density state. On the other hand, the spherical particles, even at the close packing achieved in the numerical simulations, may not establish an effective ground arch, so that the effect of the settlement of the trapdoor could propagate up through the sand layer to the surface above.

Here, we have verified that the earth pressure on the trapdoor and the surroundings could be modelled using DEM methods. On the other hand, the surface settlement could not be evaluated using a simple calculation in the DEM analysis. In near future, we would like to extend the DEM modelling to include the stabilisation of the tunnel face using shotcrete, rock-bolts, and steel arch supports and also improve the modelling of ground surface settlement.

3.2 Gravity flow tests

The above confirms that the DEM analysis expresses well the vertical stress as the trapdoor lowered; although the settlement of the surface of the sand above the trapdoor is least satisfactorily modeled. Here, we wish to investigate the distribution of the vertical stress on the floor adjacent to the position of the trapdoor during the gravity flow following the deletion of the trapdoor. This might give some insight into the processes following a small collapse of a part of the roof of a tunnel. After small collapse, the stresses at the foot of the ground arch may increase to such an extent that total collapse occurs.

Figure 5 shows the vertical stress against elapsed time after the trapdoor was removed. The elapsed time of 0.4sec corresponds with the displacement of the trapdoor of 2.0mm in the Figure 2. The vertical stress at element A undergoes a rapid cyclic variation suggesting that in the DEM simulations the establishment of the ground arch is a complex dynamic process rather than a static one. The vertical stresses on elements B and C were not much changed from the initial vertical stress calculated when the bed of spherical particles had come to equilibrium.

Figure 6 shows the vertical stress against the horizontal distance from the centre of the trapdoor at an elapsed time of 0.3sec after the trapdoor was removed. The elapsed time of 0.3sec corresponds with the displacement of the trapdoor of 1.5mm in the Figure 2. Note that the vertical stress at element A, just next to the trapdoor during gravity flow was 2.26 times that calculated at the displacement of the trapdoor of 1.5mm when the trapdoor was lowered at a steady rate. This is evidence of the increased stresses at the foot of the ground arch mentioned in the Introduction.

3.3 Implications for tunnel construction

With regard to tunnel construction it is possible to draw two conclusions from the experiments of Kikumoto et al (2003) and the DEM modeling discussed in this paper. First, the steady lowering of the trapdoor indicates that yielding of parts of a tunnel lining system is unlikely to generate large increases in loading on adjacent parts of the lining. Second, sudden collapses of part of the tunnel face may induce dynamic effects which lead to large increases in loads on other parts of the system. This in turn provides some indication for the effectiveness of the New Austrian Tunnelling Method, that is the immediate placement of support provided by the NATM even before it is fully stiffened, is effective because it prevents even partial collapses.

4 CONCLUSIONS

We performed DEM analysis of a bed of sand modeled using spherical particles in order to investigate how the distribution of the vertical stress on the supporting lower boundary of the sand container changed during steady lowering of the trapdoor and during gravity flow following the sudden removal of the trapdoor. The summary of the results obtained from this work is as follows: 1) The DEM analysis modeled well the changes in vertical

stress during lowering of the trapdoor, Figures 2 and 3. 2) The DEM calculated settlement of the surface of the sand

above the trapdoor severely over-predicted the experimentally observed values, Figure 4. Our suggested explanation for this difference is that the relatively large spherical particles used in the DEM modeling do not represent adequately the dilatancy properties of Toyoura sand.

3) During gravity flow, after the sudden removal of the trapdoor, the vertical stress on the floor immediately adjacent showed a complex dynamic variation. With increasing lateral distance from the opening this variation was much less significant, Figure 5.

4) The maximum vertical stresses on the floor next to the opening, after the sudden removal of the trapdoor, were several times larger than the maxima when the trapdoor was lowered at a steady rate, Figure 6.

5) In both the steady lowering of the trapdoor and the gravity flow after the sudden removal, the lateral distribution of the vertical stress on the floor calculated with the DEM software exhibited a saw-tooth variation (Figures 3 and 6).

The implications of this DEM modelling for tunnel construction is that even a yielding support system, such as shotcreting applied to a tunnel heading immediately after excavation, is very significant because it protects against large dynamic pressure that could be induced during a partial collapse.

5 REFERENCES

Costa, Y. D., Zornberg, J. G., Bueno B. S. and Costa C. L. 2009. Failure mechanisms in sand over a deep active trapdoor, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 1741-1753.

Hori, T., Kikkawa, N., Okita, T. and Mitachi, T. 2010. Measurement of S-wave and P-wave velocities by Bender Element tests, Proc. of the 6th conference of the Kanto Branch of the Japanese Geotechnical Society (GeoKanto2010), 4p (in Japanese).

Itasca Consulting Group Inc. 2012. Particle Flow Code in 3 Dimensions Theory and Background.

Kikumoto, M. & Kishida, K. 2003. Mechanical behavior on the sandy ground through the 3-D trapdoor experiment, Proc. of the 12th

Asian Regional Conference on Soil Mechanics & Geotechnical Engineering, 4p.

Kikumoto, M., Kimura, M., Kishida, K. and Adachi, T. 2003. Three dimensional trapdoor experiments and its numerical analyses on the mechanical behavior during tunnel excavation, Journal of Geotechnical Engineering (III), Journal of Japan Society of Civil Engineers, Vol. 65, No. 750, 145-158 (in Japanese).

Kiyama H. and Fujimura H. 1983. Application of Cundall’s discrete block method to gravity flow analysis of rock-like granular materials, Journal of Japan Society of Civil Engineers, No. 187, pp. 95-108 (in Japanese).

Murayama S. and Matsuoka H. 1971. Earth pressure on tunnels in sandy ground, Journal of Japan Society of Civil Engineers, No. 333, pp. 137-146 (in Japanese).

Rabcewicz, L. V. 1964a, b, 1965. The New Austrian Tunneling Method, part one, two and three, Water Power, November, December and January, 453-457, 511-515 and 19-24.

Terzaghi, K. 1936. Stress distribution in dry and saturated sand above a yielding trap-door, Proc. 1st International Conference on Soil Mechanics and Foundation Engineering, Cambridge, Mass., 35-39.

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Microstructural changes leading to chemically enhanced drainage

Modifications de microstructure entraînant un drainage chimiquement amélioré

Minder P., Puzrin A.M. ETH Zurich, Institute for Geotechnical Engineering, Zurich, Switzerland

ABSTRACT: The hydration state of clay mineral surfaces is a key influence factor on the mechanical and hydraulic behaviour of clays. Small changes of the cation occupancy of these surfaces can lead to pronounced changes in macroscopic material parameters.The sensitivity of the material response allows for designing chemical soil improvement by selectively exchanging the cations. In this study we explore the effect of a targeted cation exchange in smectite clays to modify soil properties in situ. The highly selective and strongly exchanging organic cation guanidinium was used to stabilise the interlayer distance between clay platelets.

On the particle scale the cation exchange led to the formation of stable aggregates. Mercury intrusion porosimetry and oedometrictests confirmed the stability of these aggregates and of an opened pore structure also under high stresses. Macroscopically, the modification resulted in a permanently increased permeability. The magnitude of the improvement is such, that after infiltration of thechemical into a clayey soil around an injection pipe, the modified soil zone could act as a drainage conduit. Potential application ofsuch flexible drainage systems are in creeping landslides, where continuing displacements cause failure of conventional drains.

RÉSUMÉ : Un facteur d’influence prédominant pour le comportement mécanique et hydraulique des argiles est l’état d’hydratation des surfaces des minéraux argileux. La sensibilité accentuée de la réponse du matériau permet de concevoir des méthodesd’amélioration chimiques pour sols au moyen d’un échange ciblé de cations. Dans cette étude, nous explorons les effets d’un teléchange dans le contexte d’un argile smectique pour modifier les caractéristiques d’un sol in situ. Le cation organique guanidinium a été utilisé pour stabiliser l’espace entre les couches des lamelles d’argile. Au niveau particulaire, l’échange de cations a engendré desagrégats stables. Les essais oedometriques et de porosimétrie par intrusion de mercure ont confirmé la stabilité de ces agrégats et une structure de pore ouverte, même sous forte sollicitation. Au niveau macroscopique, la modification se manifesta par une augmentationpermanente de la conductivité hydraulique. L’amélioration est telle qu’après l’infiltration du produit chimique dans un sol argileux entourant une conduite d’injection, le sol affecté peut être utilisé comme drain. Les applications de tels drainages se trouvent parexemple dans les glissements de terrains en état de fluage, où les déplacements continuels entravent le fonctionnement de drainages conventionnels.

KEYWORDS: Soil improvement, chemical modification, hydraulic conductivity, clay minerals, drainage.

1 INTRODUCTION

Seepage of groundwater in creeping landslides is a key parameter for the creep velocity. A reduction of the water table by means of drainage could increase the overall safety of critical slopes. Conventional drainage systems based on rigid drainage pipes are prone to failure due to the on-going deformation in such unstable areas. The development of alternative drainage techniques with increased operating life in creeping landslides has therefore a high potential in commercial application.

For fine grained soils, both theoretical and experimental studies have pointed out a high dependency of the permeability on the pore fluid. Major influence factors are the type of liquid (mainly via its dielectric constant, Fernandez and Quigley 1985), type of dissolved salts and ionic strength of the solutes (e.g. Madsen and Mitchell 1989, Lagaly et al. 2006). These studies were often carried out in the context of hydraulic barrier design and containment of nuclear wastes, where the increase was an unintended and dangerous effect caused by contaminants and leachates.

With the goal of purposely creating zones of higher permeability in-situ as part of a drainage system, this paper focuses on the fundamental aspects required in the development of an innovative soil improvement technique. Enhancing soil permeability with chemicals may be accompanied with unexpected side effects such as deterioration of stiffness or strength. In field applications such mechanical consequences could result in excessive deformation or failure. The effects on

other geotechnical parameters than hydraulic conductivity therefore need to be addressed as well.

2 MATERIALS AND METHODS

Mineralogical investigations indicated that guanidinium cations affect directly the interlayer distance of the stacked sheet-silicate structure of montmorillonite (Plötze and Kahr 2008). This strongly binding cation is therefore chosen as a possible chemical agent to increase permeability by inhibition of interlayer swelling.

In order to assess the mechanisms behind the increase in permeability caused by guanidinium, experiments on different scales were performed. To quantify the changes in soil fabric and structure due to the cation exchange, a closer look on the resulting changes on particle and aggregate scale was taken. The macroscopic stability of the new features was subsequently determined in standard geotechnical tests.

2.1 Materials

The laboratory tests were carried out on soil samples reconstituted with commercially available, standardised constituents. A commercial Ca-bentonite (Calcigel, Südchemie, Germany) with a total montmorillonite content of 65% was used as fine grained component. Other occurring mineral phases in this bentonite were quartz, feldspar, kaolinite and mica. Where

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appropriate the bentonite was stretched with rounded inert quartz grains (washed Perth sand, d50 = 0.24 mm, Cook Industrial Minerals, Australia).

The pore water of the reconstituted samples was prepared with a 0.01 mol/L CaCl2 solution as standardised groundwater equivalent.

Guanidinium solutions were prepared from analytical grade guanidinium hydrochloride salt (C(NH2)3Cl, ≥99%, Fluka Analytical, Switzerland) and demineralised water.

2.2 Investigations on particle scale

Homo-ionic bentonites were produced by saturation of Calcigel-clay with 1.0 mol/L calcium and guanidinium solutions, respectively. Images of the particles and aggregates were taken with a scanning electron microscope. The grain size distribution of the suspended material was measured with a Laser scattering analyser (Partica LA-950, Horiba, Germany).

After saturation, the modified soils were washed in suspension with demineralised water to remove excess ions and to avoid salt precipitation in the inter-particle pore space. Compacted samples were reconstituted from the washed material at water contents slightly above to their liquid limit. Mercury intrusion porosimetry (MIP, PASCAL 240/440, Porotec, Germany) on dry samples was used to quantify the alterations of the pores system due to guanidine treatment. Crack-free pieces of the slowly dried clays were subjected to vacuum evacuation for 2 h prior to mercury intrusion.

2.3 Macroscopic tests

The primary and intended soil improvement of chemically enhanced drainage is the increase in hydraulic conductivity. This increase was measured based on the evaluation of time settlement curves on bentonite samples. For this purpose, compacted samples of unmodified bentonite were reconstituted and mounted into a standard oedometer cell. The chemical agent was then delivered to the soil in the pore water by diffusion from the top and bottom filter plates. Subsequent loading up to 800 kPa allowed deriving hydraulic conductivities at different void ratios. Untreated samples served as reference for comparison.

Oedometric tests on compacted mixtures of bentonite (40 %) and quartz sand (60 %) were carried out to assess both the increase in permeability and the effects on stiffness due to the chemical treatment. Samples were reconstituted with different chemical composition of the pore water - either artificial groundwater or guanidinium solutions - and subsequently tested according to the procedure given in ASTM D2435-04.

The same mixtures were used in constant head permeameter tests, were the permeability was measured on both modified and unmodified soil. Additionally, the temporal evolution of the permeability during flow-through treatment with guanidinium solutions was recorded for a sample with initially unmodified soil.

Since chemically enhanced drainage is planned to be applied in the context of stabilisation measures, the effects of the chemical modification on the strength parameters of the soil should not be neglected. Therefore, we assessed the influence on the residual shear resistance of pure bentonite samples with a ring shear apparatus. The samples were directly reconstituted from homo-ionic calcium and guanidinium bentonites, as the device used in this study did not allow for chemical modification of the soil within the sample cell.


3.1 Modification of particles and pores

During sample preparation of modified bentonites, a granular, non-plastic behaviour was observed. Images taken with a

scanning electron microscope (Figure 1) revealed that in suspension the clay fraction aggregated upon addition of guanidinium.

The measurement of the particle size distribution supported this observation. Both calcium and guanidinium bentonites feature a bimodal distribution. However, figure 2 shows that the total volume fraction of the larger mode – containing the aggregates – is almost doubled for the guanidinium samples (42.4 %) compared to the calcium clay (22.4 %).

Figure 1. SEM-images of bentonite grains after washing in suspension with demineralised water. The calcium form remains finely dispersed (left), whereas the exposure to guanidinium ions (right) leads to the formation aggregates.

Figure 2. Bimodal particle size distribution measured with laser diffraction. The volume fraction of the larger mode (aggregates) is significantly increased by the chemical modification.

Compacted samples were analysed with MIP in order to examine whether these aggregates were capable of maintaining an open pore structure. Considerable changes in the pore system were detected. Even though the unmodified soil was prepared at a higher water content (wL,Ca = 102%) the modified bentonite (wL,Gnd = 64%) features a larger accessible pore volume (Figure 3). The largest contribution to the additional pore volume stems from pores with average radii of 2 µm. The total pore volume of these larger pores had increased, the volume in the smaller pore fraction (radius < 0.1 µm) however was slightly reduced compared to the reference material.

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Figure 3. Cumulative pore size distribution of two bentonite samples measured with mercury intrusion porosimetry. The modified material features a much larger pore volume fraction in pores with radii of about 2 µm.

3.2 Geotechnical parameters

The tests on pure Calcigel samples, where the chemical was delivered without mixing but diffusively, showed a constantly improved hydraulic conductivity even with this gentle treatment method. The evaluation of the time-settlement curves based on the theory for one dimensional consolidation showed that – although the absolute permeability decreased with increasing compaction – the relative improvement compared to the raw material was preserved and ranging in the order of one magnitude (Figure 4). The aggregated clay minerals are apparently able to keep the newly created flow paths open also without the presence of a rigid sand skeleton.

Oedometer tests on a triplicates series of the quartz/bentonite mixtures were additionally analysed with respect to changes in stiffness. The difference between unmodified and natural soil was small and in the order of the standard deviation for both initial loading and unloading/reloading (Table 1). The evaluation of the time-settlement curve again provided hydraulic conductivities at different stress levels. Figure 5 clearly shows, that despite vertical stresses of up to 800 kPa (corresponding to the smallest void ratios), the increased permeability is maintained during compaction.

Table 1. Stiffness parameters averaged over three samples and three oad steps each. Standard deviation is given in parenthesis. l

Raw Material Modified Material

Compression index Cc 0.38 (0.04) 0.35 (0.02)

Swelling index Cs 0.09 (0.03) 0.11 (0.03)

By means of permeameter tests the magnitude of possible

improvement under flow-through conditions was determined. By mixing soils with guanidinium solutions an increase of the permeability by the factor 30 was achieved (Figure 6). Even when unmodified samples were simply permeated with guanidinium solutions – instead of water – the average hydraulic conductivity increased by one order of magnitude. In potential field application this method of treatment could facilitate the delivery of the chemical in-situ without mechanical disturbance.

Figure 4. Decrease of hydraulic conductivity during sample compaction (including log-linear regression) of bentonite samples. The relative improvement is not deteriorated during compaction.

Figure 5. Decrease of hydraulic conductivity during sample compaction (including log-linear regression) of quartz-bentonite mixtures. For identical void ratio the modified soil is constantly about one order of magnitude more permeable.

Figure 6. Evolution of hydraulic conductivity during flow-through treatment (dotted line) with guanidinium solution. For comparison the data for raw material (dashed line) and reconstituted modified soil samples (full line) permeated with water are plotted as well.

The effects of the chemical modification on strength parameters are presented based on ring shear tests. The residual

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shear resistance was measured by a multi-stage procedure with a constant shear velocity of 0.2 mm/min and three different load steps for each sample. Figure 7 shows the residual shear stress after 30 to 40 mm of displacement. Moreover, the first load step was used to measure one value for the peak shear resistance. By assuming zero cohesion this peak value was used to derive the peak friction angle for each material. Table 2 summarises the friction angles obtained from this data set. It shows that both peak and residual resistance are increased after chemical treatment.

Table 2. Strength parameters of bentonite samples. For all values a rictional material with zero cohesion was assumed. f

Raw Material Modified Material

Peak friction angle 24° 31°

Residual friction angle 7° 11°

Figure 7. Results of multi stage ring shear tests on bentonite samples. The empty symbols represent peak shear resistance of the initial failure during the first load step loading. Residual state (full symbols) was reached after 30 to 40 mm of displacement in each load step.

4 CONCLUSIONS

Based on the experimental work on different scales the following conclusions could be drawn:

The molecular change of the clay minerals on the level of the surface chemistry results on the particle scale in an aggregation of small clay minerals to larger and stable particles. The aggregated, non-swelling particles provide an open pores system with pores of about 2 µm. Upon wetting these pores are no longer filled with an expanding clay phase but kept open. The pore water is not bound as immobile interlayer water and remains available for circulation.

The compression tests have shown that the new structure is stable also under stresses up to 800 kPa for both pure bentonites and quartz/bentonite mixtures. Despite these drastic changes on the microstructural level, the stiffness is not significantly affected by chemical treatment. The material strength is positively influenced by the chemical treatment as both peak shear resistance and residual shear resistance for the modified bentonite is higher compared to the raw material. A negative influence on these mechanical properties can therefore be excluded.

In summary, the findings are positive indicators for further development of the technique towards purposely creating zones

of higher permeability in-situ as part of a drainage system in creeping landslides.

5 ACKNOWLEDGEMENTS

The authors would like to thank Dr. M. Plötze of the ClayLab at ETH Zurich for his help during chemical laboratory testing. This work has been supported by grant Nr. 200021-137689 from the Swiss National Science Foundation, Switzerland.

6 REFERENCES

ASTM Standard D 2435, 2004. One-Dimensional Consolidation Properties of Soils Using Incremental Loading, ASTM International, West Conshohocken, USA.

F. Fernandez, R.M. Quigley, 1985. Hydraulic conductivity of natural clays permeated with simple liquid hydrocarbons. Can. Geotech. J., 22(2), 205-214.

G. Lagaly, M. Ogawa, I. Dékány, 2006. Clay Mineral Organic Interactions. In: F. Bergaya, B.K.G. Theng, G. Lagaly, (eds.) Handbook of Clay Science, Elsevier, 309-377.

F.T. Madsen, J.K. Mitchell, 1989. Chemical effects on clay hydraulic conductivity and their determination. Mitteilungen des IGBM, ETH Zürich, 135, 67.

M. Plötze, G. Kahr, 2008. Diagnostic intercalation in clay minerals – use of Guanidine carbonate. Proc. of the 4th MECC, Mineralogia, 33, 132.

1031

Discrete Element Method Study of Shear Wave Propagation in Granular Soil

Étude de la propagation des ondes de cisaillement dans un sol granuleux par la méthode des éléments discrets

Ning Z. Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, NC, USA

Evans T.M. School of Civil & Construction Engineering, Oregon State University, Corvallis, OR, USA

ABSTRACT: Shear wave velocity is a fundamental parameter for describing the small-strain response of soil and is a critical input for multiple constitutive models used to describe the static and dynamic behavior of granular materials. Shear wave velocity is understoodto be a function of particle parameters (shape, elastic properties, gradation) and state (void ratio, boundary stress), but the exact effectsof these parameters are difficult to measure using laboratory results alone. This paper presents results of a study of shear wavepropagation in granular soil using the discrete element method (DEM). In this study, cylindrical assemblies of particles were subjected to shear wave excitation at one end and axial propagation velocities were measured. The effects of excitation frequency,particle size, and confining stress were investigated. Micromechanical observation of the specimen is presented and analyzed in terms of particle velocity vectors.

RÉSUMÉ : La célérité de l’onde de cisaillement est un paramètre fondamental pour décrire le modèle constitutif du sol. Plusieurs modèles utilisent ce paramètre pour décrire le comportement statique et dynamique de matériaux granuleux. La vitesse de l’onde decisaillement est fonction des paramètres des particules (la forme, les propriétés élastiques, la granularité) et de leur état (indice des vides, contrainte aux limites). Par contre, il est difficile de mesurer l’effet exact de ces paramètres en utilisant des résultatsexpérimentaux. Cet article présente les résultats de l’étude de la propagation de l’onde de cisaillement dans un sol granuleux en utilisant la méthode des éléments discrets. Les particules, groupées en cylindre, sont excitées par des ondes de cisaillement à un boutdu cylindre. Les vitesses de propagation sont ensuite mesurées. Les effets de la fréquence d’excitation, de la taille des particules et de la contrainte de confinement sont étudiés. Les observations du spécimen sont présentées et analysées en termes de vecteurs de céléritédes particules.

KEYWORDS: small strain, shear wave velocity, discrete element method. 1 INTRODUCTION

Shear (S-) wave velocity is a fundamental parameter for describing the small-strain response of soil and is a critical input for multiple constitutive models used to describe the static and dynamic behavior of granular materials (Santamarina 2001). S-wave velocity is understood to be a function of particle parameters (e.g. shape, elastic properties, gradation) (Patel et al. 2009) and state (e.g. void ratio, boundary stress) (Hardin and Richart 1963). Many of these properties can be measured (or observed) as specimen-averaged quantities, but in some cases, the parameter of interest is not directly observable using standard laboratory practices. For example, Agnolin and Roux (2007) have shown that shear wave velocity is a function of both void ratio (packing fraction) and coordination number (i.e., specimens with identical void ratios but different coordination numbers exhibit different wave propagation speeds). This complex behavior implies a need for investigation of wave propagation in particulate assemblies that goes beyond traditional specimen-averaged approaches. Discrete element method (DEM) simulations are a useful tool for investigating the complex behavior of particulate materials in conjunction with laboratory tests. In terms of wave propagation, 2D DEM simulations have been conducted to study the general relationships between wave propagation variables and soil fabric (Sadd et al. 1993). In a DEM study of the acoustic properties of weakly cemented sandstone by Li and Holt (2002), a logic similar to S-wave generation and measurement in laboratory tests was applied. O’Donovan et al.

(2012) recently used 2D DEM models to simulate bender element tests on an idealized granular material. In the current study, 3D DEM simulations were used to simulate S-wave propagation in a granular material. The effects of excitation frequency, particle size and confining stress are investigated and compared to published trends of small-strain response for granular materials. Micromechanical observation of the specimen is presented in terms of particle velocity vectors.

2 SIMULATION OF SHEAR WAVE PROPAGATION

2.1 Generation of DEM assembly and shear waves

Cylindrical DEM specimens were generated with the following properties: Gs=2.7, D50=2.0 mm, Cu=1.2, Gg=2.9 GPa, νg=0.31, and μ=0.31 (note that Gg, νg, and μ are grain, not specimen, parameters). Two planar rigid walls defined the top and bottom boundaries of the specimen and were used to control the applied vertical stress. Radial confinement was supplied by stacked cylindrical walls (Zhao and Evans, 2009) to simulate a flexible membrane, which can help to minimize the wave reflection from the lateral boundaries (O’Donovan et al. 2012). S-waves were generated by applying a horizontal excitation to a thin layer of particles at one end of the specimen using a sinusoidal pulse. Compared to the S-wave generation method used in the bender element test, in which the wave source is a point and the wave propagation front is spherical, the S-wave generation method in the current study can help to reduce the compression (P) wave interference effect (Lee and Santamarina

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2005) observed in physical specimens. In DEM simulations the displacement and velocity of each individual particle can be monitored, allowing for the specification of multiple wave receivers at arbitrary locations. Figure 1 shows five particles along the central axis of the cylindrical specimen that were selected as S-wave receivers.

Figure 1. A DEM specimen with S-wave transmitting layer and receivers (for clarity, only half of the specimen is shown).

2.2 Interpretation of travel time

Accurate determination of signal travel time has been the subject of considerable research in laboratory S-wave velocity measurements (e.g., Styler and Howie 2012). Many factors, such as cross-talk between the source and the receiver (Lee and Santamarina 2005), system delay (Yang and Gu 2012), fabrication defects in the testing device (Montoya et al. 2012), and electric and environmental noise add to the uncertainties and the difficulties in the interpretation of receiving signal.

In DEM simulations, most of these aforementioned influence factors can be eliminated. Figure 2 shows ‘clean’ source and receiving signals from a typical DEM S-wave propagation simulation. The amplitude of the signal is a representation of the displacement of particles in the direction of the excitation (along x-axis in Figure 2). It can be seen that the receivers’ respond to the excitation at different times. The delays of the first arrival between equally spaced receivers are almost the same. The receiving signals have identical wave forms but attenuate as the distance to the source increases. In laboratory, S-wave velocity is typically calculated from the travel time and the distance between wave transmitter and receiver, while in DEM simulations it is also possible to calculate S-wave velocity between the receivers. The common start-to-start and peak-to-peak methods can be applied between source and receiver and between any two receivers. In simulation, when applying the cross-correlation method (Viggiani and Atkinson 1995) between receiving signals, it is expected to be ‘cleaner’ to interpret compared to laboratory data because they contain less noise and there is a very high similarity between the waveforms. In the bender element test, the wave motion is indirectly expressed in the change of voltage, which means that the initial polarization between the input and the output signal does not necessarily reflect the relative direction of wave motions at the wave source and at the receiver. It is for the reason that the polarity of the signal is not only determined by the direction that the bender element curves but also affected by the wiring of the bender element electrodes. Unlike in the bender element test, the polarity of the signal in the current simulations directly represents the direction of wave motion, which helps to identify whether the initial deflection of the receiving signal is caused by the S-wave (which results in the same polarity as the source

signal) or by the P-wave reflected from the side boundaries (which results in an inverse polarity of the source signal). From Figure 2, it can be seen that the initial deflection of the receiving signals has the same polarity as the source signal, which indicates little P-wave interference to the first arrival of the receiving signal. The S-wave velocities shown in this manuscript were determined as follows: the travel times between the source layer and the two far most receivers (to avoid near-field effects; Sanchez-Salinero et al. 1986) were determined by the start-to-start method; second, the final representative S-wave velocity was determined by averaging the travel times obtained in step one. S-wave transmitting layer

Figure 2. Source signal and receiving signals from a typical DEM S-wave propagation simulation (D50 = 2.0 mm; e = 0.63; σ'3 = 150 kPa. Note: for clarity, the amplitudes of the receiving signals are upscaled to facilitate comparison)

3 EFFECTS OF EXCITATION FREQUENCY

In bender element tests, the response of the receiver bender element is enhanced when the frequency of the input signal approaches the resonant frequency of the bender element-soil system (Lee and Santamarina 2005). The resonant frequency can be determined in the laboratory by sweeping the excitation frequency of a sine pulse. The DEM models were excited by sine pulse with different frequencies to approximately identify the resonant frequency. Figure 3 shows the response from one of the receivers in a DEM specimen excited by sine pulse signals with a frequency range from 1 kHz to 100 kHz. From Figure 3 it is apparent that the response of the receiver is rather weak under excitation frequencies of 1 kHz and 2 kHz. The strongest response is achieved at 5 kHz. Thus, the resonant frequency is located in the range 2-10 kHz. This agrees with published laboratory findings (Santamarina and Fam 1997). The response signals are strong with similar waveform across a wide range of excitation frequencies from 5 kHz to 100 kHz. Frequency domain analyses were also used to investigate the effect of excitation frequency (Figure 4). It can be seen that the highest amplitude occurs at around 3 kHz, consistent with the observations from time domain analyses. Figure 4 also shows that the receiving signal contains a range of frequency components though the transmitted signal is a sine pulse. When the frequency of the transmitted signal is lower than the resonant frequency (3 kHz), the predominant frequency of the corresponding receiving signal is identical to the frequency of the transmitted signal. However, when the frequency of the transmitted signal is higher than the resonant frequency, the predominant frequency of the corresponding receiving signal is roughly equal to the resonant frequency. The DEM specimen works as a low-pass filter, rejecting frequency components higher than the cut-off frequency, which is found to be the resonant frequency discussed above. The analytical illustration of this low-pass filtering effect in discrete media can be found

R2

R1

R3

R4

R5Receivers

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in the work by Santamarina (2001). This phenomenon was also reported by Yang and Gu (2012) in their experimental study.

Figure 3. Response of a receiver in the time domain, excited by a sine pulse with different frequencies (D50 = 2.0 mm; e = 0.63; σ'3 = 150 kPa. Note: for clarity, the scale of the magnitude for the source signal and the receiving signals are different)

Because the materials with internal spatial scales are inherently dispersive (Santamarina 2001), wave propagation velocity varies with frequency in granular soils. There are experimental studies (Blewett et al. 2000; Styler and Howie 2012) showing frequency-dependent S-wave velocity responses. In the 2D DEM simulation by O’Donovan et al. (2012), the S-wave velocity increases linearly with the transmitted frequency.

Figure 4. Response of receiver in frequency domain, excited by sine pulse with different frequencies (D50 = 2.0 mm; e = 0.63; σ'3 = 150 kPa).

Figure 3 shows that the variation of the first arrival of the S-wave can be observed, though it is not obvious. The S-wave velocity increases slightly (from 220 m/s to 231 m/s) when the transmitted frequency increases from 1 kHz to 5 kHz. This is attributed to the viscous damping effect at the inter-particle contacts (O’Donovan et al. 2012). When the transmitted frequency is higher than the resonant frequency, the variation of S-wave velocity becomes even less appreciable (a consequence explained by the aforementioned low-pass filter effect).

4 PARTICLE SIZE AND CONFINING STRESS EFFECTS

The effect of particle size on S-wave velocity has been widely studied by using bender element tests. A recent work by Yang and Gu (2012) found controversial results by comparing many previous researches. DEM simulation allows for the study of particle size effect with a much larger size range than physical test does. Knowing the particle size effects on wave propagation problem is also important for DEM simulation to determine whether the mass-scaling (O'Sullivan, 2011) is applicable. Three mean particle sizes were considered (2 mm, 20 mm, 200 mm). Since wave propagation involves high frequency effects, different responses are expected from models with different particle sizes. Figure 5 shows the effects of particle size on S-wave velocity and resonant frequency. There was little change in S-wave velocity over three orders of magnitude in particle

size. This agrees with Yang and Gu (2012), who found that S-wave is effectively size independent. Regarding the resonant frequency of model, a linearly decreasing trend with the particle size was observed.

Figure 5. Effects of particle size on S-wave velocity and resonant frequency (D50 = 2 mm, 20 mm, 200 mm; e = 0.62–0.63; σ'3 = 150 kPa)

These results indicate that mass-scaling (e.g. by manipulating the particle size; Evans and Frost 2007, Jacobson et al. 2007, Belheine et el. 2010) can be applied to reduce computing time in DEM simulations of S-wave propagation. The excitation frequency should be carefully selected near the resonant frequency (a function of the particle size) to obtain strong frequency response in the model. Stress state affects interparticle stiffness (Santamarina 2001) and, perhaps more significantly, contact quality (Evans et al. 2011) and thus, wave propagation speed. Many empirical relationships between S-wave velocity and effective confining stress have been proposed (Hardin and Richart 1963) for sands. One general form is as follows:

(1) where α and β are fitting parameters and σ' is the effective confining stress in kPa. In this study, the S-wave velocities of a DEM specimen with D50 = 2.0 mm were determined under confining stresses ranging from 50 to 900 kPa. The simulation results present a similar trend as observed in the lab as shown in Figure 6. The fitting parameters α and β were found to be 95.5 and 0.18 respectively, which fall into the range of typical values for sand and OC clay (Fernandez 2000)

Figure 6. Effects of confining stress on S-wave velocity (D50 = 2.0 mm; e = 0.62-0.63) 5 MICROMECHANICAL OBSERVATIONS

In laboratory tests, it is not possible to directly observe complex wave motions within the specimen. DEM simulations allow for micromechanical predictions of material response, which can help to provide a better understand of the complexity of wave propagation mechanisms in granular materials. Observations of the particle velocity vectors are briefly considered below. Figure 7 shows particle velocity vectors on three specific cutting planes of a DEM specimen 10 ms after excitation. The cutting plane on the left goes through the central axis of the specimen with its normal parallel to Y-axis. The cutting plane 1-1 and the cutting plane 2-2 are at the height of two receivers respectively with their normal parallel to Z-axis. The particle

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velocity vectors on these cutting planes show dominant S-wave motions (from right to left) in the central part of the specimen while minor P-wave motions developed on the sides (with the particle on the right moving downwards and the particle on the left moving upwards). Though the P-waves travel faster, they cause little interference at the central part of the specimen where the receivers are located. At the time shown in Figure 7, the S-wave front just passes cutting plane 1-1 but has not yet reached cutting plane 2-2. A significant amount of additional particle level information is available from DEM simulations, including displacement vectors, contact forces, and contact slip. This information is being analyzed as part of ongoing studies into the fundamental nature of shear wave propagation in granular assemblies.

7 REFERENCES

Agnolin, I., and Roux, J. (2007). "Internal states of model isotropic granular packings. III. Elastic properties." Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 76(6).

Belheine, N., Plassiard, J. -., Donze, F. -., Darve, F., and Seridi, A. (2009). "Numerical simulation of drained triaxial test using 3D discrete element modeling." Comput.Geotech., 36(1-2), 320-331.

Blewett, J., Blewett, I. J., and Woodward, P. K. (2000). "Phase and amplitude responses associated with the measurement of shear-wave velocity in sand by bender elements." Can. Geotech. J., 37 1348-1357.

Evans, T. M., and Frost, J. D. (2007). "Shear banding and microstructure evolution in 2D numerical experiments." Geo-Denver 2007: New Peaks in Geotechnics, Febrary 18, 2007 - Febrary 21, American Society of Civil Engineers, Denver, CO, United states, 28.

Evans, T.M., T.S. Yun, and J.R. Valdes. (2011). “Effective Thermal Conductivity in Granular Mixtures: Numerical Studies,” IS-Seoul: Fifth International Symposium on Deformation Characteristics of Geomaterials, Seoul, Korea, August 31-September 3, 6 pp. 1 1

Fam, M., and Santamarina, J. C. (1997). "A study of consolidation using mechanical and electromagnetic waves." Geotechnique, 47(2), 203-219.

Fernandez, A. L. (2000). "Tomographic Imaging the State of Stresses." PhD Dissertation, Georgia Institute of Technology, Atlanta.

2 2

Hardin, B. O., and Richart, J., F.E. (1963). "Elastic wave velocities in granular soils." ASCE -- Proceedings -- Journal of the Soil Mechanics and Foundations Division, 89 33-65.

Jacobson, D. E., Valdes, J. R., and Evans, T. M. (2007). "A numerical view into direct shear specimen size effects." Geotech Test J, 30(6), 512-516.

Lee, J., and Santamarina, J. C. (2005). "Bender elements: Performance and signal interpretation." J.Geotech.Geoenviron.Eng., 131(9), 1063-1070.

Li, L., and Holt, R. M. (2002). "Particle scale reservoir mechanics." Oil and Gas Science and Technology, 57(5), 525-538.

Montoya, M., Gerhard, R., DeJong, J., Weil, M., Martinez, B., and Pederson, L. (2012 (in press)). "Fabrication, Operation, and Health Monitoring of Bender Elements for Aggressive Environments." Geotechnical Testing Journal, 35(5), 1-15.

Figure 7. Particle velocity vectors on different cutting planes of a DEM specimen at a 10ms time point after excitation

6 CONCLUSION

This paper presents a DEM study of S-wave propagation in random assemblies of spherical particles. DEM simulations provide high quality receving S-wave signals, given that the respsone is free of factors such as cross-talk, systen delay, and environmental noise. A multiple receiver setup allows for more reliable S-wave velocity dermination.

O'Donovan, J., O'Sullivan, C., and Marketos, G. (2012). "Two-dimensional discrete element modelling of bender element tests on an idealised granular material." Granular Matter 14, 733-747.

O'Sullivan, C. (2011). Particulate discrete element modelling : a geomechanics perspective. Taylor & Francis, London.

Excitation frequency showed significant impact on the response of specimen. Similar to laboratory observations, a resonant frequency that resulted in the strongest resposne was identified in DEM simulations. The granular spcecimen functioned as a low-pass filter when excited by a sine pulse with different frequencies. Frequency components that were higher than the resonant frequency were significanlly attenuated. Dispersion was observed when the excitation frequencies were low. The affect of excitation frequency on S-wave velocity became less appreciable when the frequencies were higher than the resonant frequency.

Patel, A., Bartake, P. P., and Singh, D. N. (2009). "An empirical relationship for determining shear wave velocity in granular materials accounting for grain morphology." Geotech Test J, 32(1), 1-10.

Sadd, M. H., Tai, Q., and Shukla, A. (1993). "Contact law effects on wave propagation in particulate materials using distinct element modeling." Int.J.Non-Linear Mech., 28(2), 251-65.

Sanchez-Salinero, I., Roesset, J. M., and Stokoe, K. H. (1986). "Analytical studies of body wave propagation and attenuation." Rep. No. Report GR 86-15,University of Texas, Austin.

Santamarina, J. C. (2001). Soils and waves : particulate materials behavior, characterization and process monitoring. Wiley, New York.

Particle size had less of an impact on S-wave veloicty while the resonant frequency presented a inverse linear relathiship with particle size. Mass scaling can be applied in DEM wave-propagation simulation in terms of S-wave velocity analysis when the transmitted frequency is selected appropriately.

Styler, M. A., and Howie, J. A. (2012). "Comparing frequency and time domain interpretations of bender element shear wave velocities." GeoCongress 2012, ASCE, Oakland, CA, 2207-2216.

Viggiani, G., and Atkinson, J. H. (1995). "Interpretation of bender element tests." Geotechnique, 45(1), 149-154.

Yang, J., and Gu, X. Q. (2012). "Shear stiffness of granular material at small strains: does it depend on grain size." Géotechnique, in press.

The S-wave velocity increased with increasing effective confining stress, and is describled by an emprical fomula previously developed based on laboratory tests. The fitting parameters obtained in this DEM simulation were similar to those for sands and OC clays measured in the lab.

Zhao, X., and Evans, T. M. (2009). "Discrete simulations of laboratory loading conditions." International Journal of Geomechanics, 9(4), 169-178.

Velocity vectors highlighted the complex motions of individual particles during wave propagation. They showed dominant S-wave motion along the central area of the specimen along with minor P-wave motion on the sides.

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Microscopic observation on compacted sandy soil using micro-focus X-ray CT

Observation microscopique par micro-tomographie à rayons X de sables compactés

Otani J., Mukunoki T. Graduate School of Science and Technology, Kumamoto University, Kumamoto, Japan

Takano D. Geotechnical Engineering Field, Port and Airport Research Institute, Yokosuka, Japan

Chevalier B. Université Blaise Pascal, Clemont-Ferrand, France

ABSTRACT: It has been suggested based on the technical report of the damages on the recent heavy rains orliquefaction after earthquakes that there are some numbers of local areas where the compaction of soils is not enoughand thus, the revision of the current compaction control has been expected. In this paper, the precise behavior ormechanism of soil compaction was discussed using the results of micro-focus X-ray Computed Tomography (CT). A series of model compaction test with sandy soils were conducted and this compacted soil was scanned at different step ofcompactions. Here, not only visualization of the behavior using CT images but also more quantitative discussion such as the spatial distribution of porosity in the soils were on the discussion after conducting image analysis of the CT images.Finally, it is believed that all those findings are valuable information for reconsidering the way of compaction control forthe riverbank.

RÉSUMÉ : Les récents rapports techniques relatifs aux dommages consécutifs aux fortes précipitations ou à la liquéfaction post-séisme suggèrent qu'il existe un certain nombre de zones localisées où le compactage des sols n'a pas été suffisant. Suite à ce constat,une reconsidération des méthodes actuelles de contrôle de compactage est attendue. Dans cet article, les comportements etmécanismes exacts du compactage des sols sont discutés à partir de résultats issus de la micro-tomographie à rayons X. Une séried'échantillons compactés de sols sableux a été réalisée puis scannée à différentes étapes du processus de compactage. Les imagesrésultant de la tomographie ont permis une analyse du processus de compactage à la fois qualitative, portant sur la visualisation desmécanismes, mais aussi quantitative, notamment vis-à-vis des variations du volume des vides dans les différents échantillons. Sur labase des résultats obtenus, une discussion sur l'évolution de la répartition spatiale des vides dans l'échantillon a également été menée.L'ensemble des résultats et observations représente une source d'information précieuse pour reconsidérer les méthodes de contrôle ducompactage des digues.

KEYWORDS: compaction, density, image analysis, sandy soil, X-ray CT.

1 INTRODUCTION

Compaction control for riverbanks is usually done by the density of the soils no matter how difference on the design criteria or soils is. However, it has been suggested based on the technical report of the damages on the recent heavy rains or liquefaction after earthquakes that there are some numbers of local areas where the soil compaction is not enough and thus, the revision of the current compaction control has been expected. Under those circumstances, the goal of this study is to develop a new quantitative compaction control for riverbank. In this paper, the soils after compaction are precisely investigated using micro-focus X-ray Computed Tomography (CT) scanner which is the one of non destructive testing methods with high resolution (Higo et. al. 2011). This investigation makes the micro level of the discussion possible.

2 SUMMARY OF TESTING

In this test, micro-focus X-ray CT (TOSCANER-32300FPD, Toshiba) at Kumamoto University was used and this apparatus makes the micro level of discussion possible in the soils without any destructions. A series of X-ray CT scanning were conducted on the model ground under the process of compaction to evaluate the change of soil property due to compaction. A sandy soil called “Yamazuna” sand was used in this study and Table 1 shows those of soil properties. As shown in Figure 1, “Yamazuna” sand has a wide range of the size of

soil particles, so that it can be suitable for soils in the riverbank.

100

80

60

40

20

00.001 0.01 0.1 1 10

Diameter (mm)

Perc

ent f

iner

by

wei

ght(

%)

Soil particle density (t/m3) 2.71

Figure 1. Particle size distribution of “Yamazuna” sand.

Table 1. Soil properties of “Yamazuna” sand.

Sand (%) 64.3

Silt (%) 4.9

Clay (%) 3.3

Uniformity coefficient 11.7

Coefficient of curvature 1.61

Classification S-F

Optimum water content (%) 12.2

Maximum dry density (t/m3) 1.90

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Table 2. Test cases and degree of compaction.

For the compaction properties, a series of compaction test were conducted and maximum dry density of 1.90 t/m3 and the optimum water content of 12.3 % were obtained. First of all, the sand was prepared in the acrylic mould (height: 100mm, diameter: 50mm) with the conditions of the optimum water contents and dry density of 1.50 t/m3. Dynamic compaction method was used with controlling its compaction energy and here, as shown in Table 2, two different cases were examined, which are Case-1 and Case-2. Case-1 is the case of relatively high energy in which the height of each time of falling rammer is 0.20 m while Case-2 is the one for low energy which is 0.10m for the falling weight. The weight of falling rammer was 9.81 N. Figure 2 shows those two cases in which the level of compaction is also shown in this figure. As shown in this figure, the amount of work was set as equal for both cases in each level of compaction although the degree of the compaction was slightly different between two cases. Table 3 shows the conditions of CT scanning and Figure 2 shows the scanning area of the specimen, which was the area from 15 mm high and up to 55mm from the bottom of the specimen and the width of scanning area was 40mm. The precise contents of X-ray CT can be found in the references (Otani 2003 and Watanabe et. al. 2012).

3 IMAGE ANALYSIS

The characteristic of compacted soils was discussed with the results of CT scanning and those are not only direct result from image visualization but also more quantitative ones such as distribution of the voids in the soil using image data. In order to obtain quantitative results, image analysis plays an important role and especially, the determination of the threshold value between two materials such as soil particles and voids is most important for this quantification. Figure 3 shows the frequency of so called “CT-value” in the whole specimen for Case-1. This “CT-value” has been known as the well correlated value with material density (Otani et. al. 2000). As shown in this figure, there are two dominant CT-values in the specimen and due to the level of compaction, those peaks are gradually changed. X-ray CT has a spatial resolution and in this case, this resolution was 75m. The sand used in this test has a fine fraction (less than 5 m) of 8.2% and as a result, there is no way to distinguish all the sizes of the particles. However, it can be said from Figure 3 that the higher peak moves to the higher frequency and the lower peak moves to the lower frequency after the compaction. This means that the increase of the CT-value due to compaction is the cause of the fact that the small particles move to the voids and then those areas are shown as the areas of higher CT-values. In the mean time, the area of low density is decreased due to the decrease of the voids. In order to discuss more quantitative sense, the threshold value of two peaks shown in Figure 3 was determined. Here, EM algorithm (Dempster et. al. 1977) which is one of the maximum likelihood methods was used and this method is useful for the case of multiple peaks of the frequency curve. Here, the calculation

CompactionEnergy (kJ/m3)

Case-1No. of Compaction

Case-1Deg. of Compaction (%)

Case-2 No. of Compaction

Case-2Deg. of Compaction (%)

Initital 0 0 79.4 0 78.3 LevelA 15 1 83.3 2 81.3 LevelB 75 5 88.2 10 84.7 LevelC 150 10 91.5 20 87.1 LevelD 374 25 97.5 50 90.6 LevelE 749 50 102.0 100 93.8 LevelF 1497 100 106.0 200 96.5 LevelG 2995 200 109.3 400 98.5

Figure 2. Area of CT scanning.

Scan area

(mm)10

040

15

500 1000 1500 2000 2500 3000

5

10

15

20

25

30

0

Compaction energy (kJ/m 3 )

Poro

sity

(%)

Case-1 measureCase-1 analysisCase-2 measureCase-2 analysis

Voltage (kV) 200-230

Current A) 350

Spatial resolution (mm) 0.075

Width of slice (mm) 0.050

Number of slices 800

FCD* (mm) 205.0

FDD** (mm) 1000

*: distance from X-ray tube to the specimen

**: distance from the specimen to the detector

Table 3. Condition of CT scanning.

Figure 4. Relationship between porosity and compaction energy.

Figure 3. Frequency of CT-value for Case-1.

0 50 100 150 200

8

2500

2

4

6

Case-1initialLevel ALevel BLevel CLevel DLevel ELevel FLevel G

(×106)

Freq

uenc

y

CT-value

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with its iteration process has to be conducted in order to obtain the optimum threshold value.


4.1 CT images

First of all, the obtained threshold value was checked with the measurement of the volume of the voids in the soil. Figure 4 shows the changes of the porosity due to the increase of compaction energy for both cases and the results of the analysis using the obtained threshold values were compared with the measurement results. As easily realized from this figure, both results are fairly close and thus, it can be said that the obtained threshold value can be effective for this study. And it is also concluded that the porosity decreases with the increase of the compaction energy. Figure 5 shows the vertical cross-sectional CT images for both cases from initial to the end of compaction level (Level G). In the CT images, the color of black indicates the area of low density while that of white shows the area of high density with total of 256 levels. It is obvious that the area of the voids decreases with increase of compaction energy for both cases. And the movement of each soil particle is integrated around the top of the specimen for both cases. Looking at the degree of compaction shown in Table 2, the Level D shows already more than 90% of compaction but there are still some areas of black color. Thus, it is concluded that average value of compaction degree, 90% has still many voids in the compacted soil.

4.2 Spatial distribution of air voids

Figure 6 shows the spatial distribution of air voids in the soil in three dimensions using the results of CT scanning. The areas of light color show those of the voids. As easily realized from

those images, the more the compaction energy is the more the porosity decreases. And when Case-1 and Case-2 are compared, it is obviously said that the porosity for Case-2 is more than that of Case-1. Especially, it is realized that the distribution of porosity in the soil is somehow different between two cases. Figure 7 shows the distribution of porosity in the soil for both cases, in which the porosity of every 5mm depth was plotted with the average values as dotted lines. As realized from those figures, it is concluded that the compaction for Case-1 which is higher energy for one cycle of compaction is more effective than that of Case-2 and especially, for Case-2, it can be realized that there are not enough compaction areas around the bottom of the soils. This means that the way of compaction changes the effectiveness of the compaction.

5 CONCLUSIONS

The main purpose of this study was to develop a new method of compaction control for riverbank. Here in this paper, micro level of the mechanism on compacted sandy soil was discussed using micro-focus X-ray CT. The conclusions drawn from this discussion are summarized as follows: (1) Based on three dimensional visualization of the compacted

soil using micro-focus X-ray CT, it has been found that the soils with 90% of average compaction degree still have some degree of the voids left; and

(2) According to the results of comparison on different compaction energy, the case of higher energy for one cycle has better compaction capacity and this was proved by the spatial distribution of the porosity in the soil.

Finally, although those findings were rather fundamental, more quantitative discussion will be continued using special image analysis such as Digital Image Correlation (DIC) method.

15 mm

(1) Case-1(a) initial (b) Level A

Figure 5. Vertical cross sectional images. CT-valueLow High

40 m

m

(c) Level B (d) Level C (e) Level D (f) Level E (g) Level F (h) Level G

(a) initial (b) Level A (c) Level B (d) Level C (e) Level D (f) Level E (g) Level F (h) Level G

(2) Case-2

1038


0 80 240

CT-valueinitial Level A Level C Level E Level G

40m

m

25mm

(a) Case-1

(b) Case-2

initial Level A Level C Level E Level G

Figure 6. Spatial distribution of porosity in the soil.

6 REFERENCES

Dempster, A.P., Laird, N.M. and Rubin, D.B. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B: Methodological, 39, 1-38.

Higo Y., Oka F., Kimoto S., Sanagawa T. and Matsushima Y. 2011. Study of strain localization and microstructural changes in partially saturated sand during triaxial tests using microfocus X-ray CT, Soils and Foundations, 51(1), 95-111.

Otani J., Mukunoki T. and Obara Y. 2000. Application of X-ray CT method for characterization of failure in soils, Soils and Foundations, 40(2), 111-118.

Otani J. 2003. State of the art report on geotechnical X-ray CT research at Kumamoto University, Keynote Lecture, X-ray CT for Geomaterials, Balkema, 43-77.

Watanabe Y., Lenoir, N., Otani J. and Nakai T. 2012. Displacement in sand under triaxial compression by tracking soil particles on X-ray CT data, Soils and Foundations, 52(2), 312-320.

0 5 10 15 20 25 30 35

20

30

40

50

Porosity (%)

Dis

tanc

e fro

m th

e bo

ttom

(mm)Air contents

initialLevel ALevel BLevel CLevel DLevel ELevel FLevel G

(a) 5mm each (b) Average initialLevel ALevel BLevel CLevel DLevel ELevel FLevel G

0 5 10 15 20 25 30 35

20

30

40

50

Porosity (%)

Dis

tanc

e fro

m th

e bo

ttom

(mm)

initialLevel ALevel BLevel CLevel DLevel ELevel FLevel G

(a) 5mm each (b) Average initialLevel ALevel BLevel CLevel DLevel ELevel FLevel G

Air contents

(a) Case-1

(b) Case-2Figure 7. Distribution of porosity in the soil.

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Study of relative permeability variation during unsteady flow in saturated reservoir rock using Lattice Boltzmann method

Étude de la variation de la perméabilité relative au cours d’écoulement transitoire dans une roche réservoir saturée en utilisant la méthode des réseaux de Boltzmann

Pak A., Sheikh B. Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

ABSTRACT: The importance of relative permeability coefficient on the productivity of oil reservoirs is well-known in PetroleumGeomechanics. Relative permeability is one of the main macroscopic parameters that heavily influence the two-phase flow regime in saturated porous rock which governs the rate of oil extraction from the well. In this study the dominant mechanisms of the flow of two immiscible fluids (water and oil) in porous media have been studied at the pore scale by using a developed simulator based on Lattice-Boltzmann Method. The validity of the numerically-derived relative permeability values demonstrate the capability of Lattice Boltzmann Method in modeling the complicated pore scale phenomena encountered in petroleum geomechanics. RÉSUMÉ: L'importance du coefficient de perméabilité relative pour la productivité des réservoirs est bien connu en géomécaniquepétrolière. La perméabilité relative est l'un des principaux paramètres macroscopiques fortement influençant le régime d'écoulement bi-phasique dans des roches poreuses saturées qui régit le l'extraction du pétrole. Dans cette étude, les mécanismes dominants del'écoulement de deux fluides non miscibles (eau et huile) dans les milieux poreux ont été étudiés à l'échelle des pores en utilisant unsimulateur développé sur la base des réseaux de Boltzmann. La validité des valeurs numériquement obtenues pour la perméabilité relative démontre la capacité de la méthode des réseaux de Boltzmann pour la modélisation des phénomènes complexes rencontrés à l'échelle des pores en géomécanique pétrolière.

KEYWORDS:Relative Permeability, Lattice Boltzmann Method, Steady/Unsteady Flow, Petroleum Geomecahnics

1 INTRODUCTION

Relative permeability is an essential petro-physical property required for description of multi-phase flow in petroleum reservoirs. It is a direct measure of the ability of the porous medium to produce one fluid when two or more fluids are present. This flow property is the result of the composite effects of porosity, pore geometry, wettability, saturation history, reservoir temperature, reservoir pressure, overburden pressure, and rock type. The relative permeability curves are very important in the study of reservoir productivity. They are used in predicting production rate and recovery from the reservoirs during all recovery stages (primary, secondary, and tertiary). Briefly, there are two basic approaches for determination of relative permeability curves from laboratory core flow tests: steady and unsteady state methods. In the steady-state method, the fluids are injected simultaneously into core plugs. In the unsteady-state method, a fluid is injected to displace another fluid present in the core. Steady-state test data processing is relatively simple, but the experiments are tedious and lengthy, because attaining steady state fluid saturations within the core requires long times, in the order of hours, following the initiation of tests under certain fluid injection rates. In contrast, unsteady-state laboratory tests can be performed rapidly and the tests better represent the real physics of the phenomenon. However, recording of a number of parameters are not possible during the experiment and also data interpretation is a much more difficult task. In both methods, data processing is further complicated unless fluid displacement rates are sufficiently high to minimize the core inlet and outlet capillary end-effects. Details on each technique are covered in Keehm et al. (2004), and Ramstad et al. (2011).

Recently, pore-scale numerical modeling has emerged for simulation of fluid flow through porous media. The main advantage of such models is incorporating the micro-scale

processes that control the large-scale phenomena. Fluid/fluid and fluid/solid interactions are examples of such processes that have significant effects on the flow regimes.

A recently developed computational fluid dynamic method which is ideal for simulating fluid flows in complicated geometries such as porous media at the pore scale is Lattice Boltzmann Method (LBM) (Chen & Doolen, 1998). LBM is suitable for modeling intricate fluid flow problems such as multiphase flow in complex structures. LBM was applied to flow through porous media soon after its emergence in 1989 (Succi, 1989). Considerable growth of its application in modeling multiphase flow through porous media mainly origins from its algorithm simplicity and accuracy in handling irregular flow paths, modeling the behavior of fluid/fluid interfaces and simulation of fluid/solid interactions (e.g. Chen & Doolen 1998; Pan et al. 2004; Schaap et al. 2007).

In this study a 2D LBM-based numerical code is developed which is capable of modeling steady-state and unsteady-state flow of two immiscible fluids through porous media. After validation of the code by some benchmark problems, a well-documented experimental work was simulated by the developed model.

2 LATTICE BOLTZMANN METHOD

The most popular LB model is the Bhatnagar–Gross–Krook (BGK) model (Chen et al. 1992) with a standard bounce-back (SBB) scheme for fluid–solid boundaries. However, some problems have encountered difficulties with this popular method. In BGK method the collision operator is approximated by a single-relaxation-time (SRT) approximation, which has some defects such as numerical instability and viscosity dependence of boundary locations, especially in under-relaxed situations (Qian et al. 1992). The viscosity dependent boundary

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conditions pose a severe problem for simulating flow through porous media because the intrinsic permeability becomes viscosity dependent, while it should be a characteristic of the physical properties of porous medium alone. The deficiencies inherent in the BGK model can be significantly reduced by using a multiple-relaxation-time (MRT) approach (He et al. 1997), which separates the relaxation times for different kinetic modes and allows tuning to improved numerical stability and accuracy. In this study we used D2Q9 MRT as they were introduced by Lallemand and Lue (2000), Extensive details can be found in Lallemand and Lue (2000), and Li et al. (2005).

The Lattice-Boltzmann method for single-phase flow describes fluid flow as collisions of mass particles in a lattice (Chen et al., 1992). In two-phase flow, we follow almost the same procedures as in the single-phase case, except that we have two different types of particles representing two immiscible fluids, and we need to calculate surface tension and wettability. There are several popular Lattice Boltzmann (LB) techniques for the analysis of multiphase flows, three of which are the methods of Gunstensen et al. (1991), Shan & Chen (1993), and free energy approach by Swift et al. (1996). All three methods have been employed in numerical researches and each one has its distinct advantages. A review of these methods can be found in Nourgaliev et al. (2003). Among all of these LBM models, Shan & Chen’s model (SC model) is widely used due to its simplicity and remarkable versatility. It can handle fluid phases with different densities, viscosities and wettabilities, and handle different equations of state as well. In this study, the multi-component (2 fluids) single phase version of SC model has been applied (Sukop &Thorne, 2006).

3 VERIFICATION

In multi-component LB models, the bubble test is often conducted to check the ability of the model in relating the pressure difference ( p), radius of curvature (R) and

interfacial tension ( ) together in the situation that a bubble of one fluid is immersed in another fluid, which should indicate that p varies linearly with respect to curvature 1/ R based on the well-known Laplace law:

(1) Four different sizes of bubbles (Figure 1) are used for the numerical experiments. Figure 1b shows the capillary pressures for four different bubbles. The theoretical prediction is shown as a solid line. The simulated values (symbols) are obtained by simply calculating pressures inside and outside the bubbles at the end of the numerical simulations. The numerical results show very good agreement with the theoretical values.

Another well-studied model of immiscible displacement, the so-called pore doublet model, is a little more complicated. A typical pore doublet consists of two tubes with different diameters, joined at both ends (Figure 2). Since the capillary pressure is inversely proportional to the radius of the tube, the capillary pressure of the smaller tube is greater than that of the bigger tube. Drainage-type snap-off occurs when the external pressure gradient is big enough to overwhelm the capillary pressure of the bigger tube, but is not big enough for the smaller tube. Theory and laboratory experiments show that under this condition the wetting phase in the smaller tube is trapped, while that in the bigger tube it is replaced by the non-wetting phase (Lenormand et al., 1983). Figure 2show that the two-phase Lattice-Boltzmann method successfully replicates the drainage-type snap-off, which tells us that the method accurately

describes capillary pressure phenomena of porous media. If the displacing fluid is wetting, then both tubes will be swept out by the wetting fluid (figure 2.b)

4 RELATIVE PERMEABILITIES

One of the most comprehensive sets of experimental works regarding relative permeability was pursued by Payatakes and his co-workers (Valavanides et al. 1998, Tsakiroglou et al. 2007) who performed experiments on the steady and unsteady flow regimes in porous core consisting of a chamber-and-throat network. Here, both steady and unsteady states experiments of Payatakes were selected for evaluation of the results of the developed LBM model. The dimensions of their specimen is 0.16 ×0.11 m, and its absolute permeability is k = 8.9 μm2. The distance between the centers of the adjacent chambers is 1221 μm, the mean throat depth is 116.6 μm, and the mean throat width is 167.5 μm.

a

b

Figure 1. (a) Four different sizes of bubbles in steady-state condition (b) capillary pressure vs. reciprocal of bubble radius. Simulated values (symbols) agree well with the theoretical prediction (solid line).

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a b

Figure 2. Drainage-type snap-off in a doublet. (a) Displacing fluid (green) is non-wetting and displaced fluid (red) is wetting. (b) Displacing fluid is wetting and displaced fluid is non-wetting.

The properties of the pair of the tested fluids are presented in Table 1. The simulations were carried out in a two-dimensional medium similar to the chamber-and-throat type networks used in the experiments (see Figure 3). It should be mentioned that the dimensions of the model could not be considered as large as that of the experiment due to high computational costs. T able 1.Physicochemical properties of fluids

Non-wetting fluid

(n-nonanol) Wetting fluid (formamide)

Viscosity(Pa s) 0.00964 0.00335

Density(kg/ ) 816 1116

Interfacial tension(mN/m) 4.3

Contact angle 9

a b

Figure 3. (a) A segment of the glass-etched chamber-and-throat network used in experiments (b) domain used in LB simulation.

At first, the saturated permeability of the medium was

determined by the numerical modeling for a steady-state Darcy’s velocity after applying a constant body force for one phase and setting the density of the other fluid equal to zero at all locations. The result was k = 8.82μm2 which is remarkably close to the experimental value of 8.9 μm2 It is important to note that the employed MRT approach in the developed LBM code has yielded more accurate predictions of both saturated as well as relative permeabilities compare to the standard BGK model, which leads to a viscosity dependent permeability.

4.1 Steady state

To simulate the steady-state experiments, we distributed fluid phases in the model according to target saturation. Flow at a given Ca is then commenced.

(2) Where is the superficial flow velocity of the injected wetting phase at the entrance, is the viscosityof the wetting

phase, is the interfacial tension.

We imposed periodic boundary conditions and allowed both fluids to enter and exit the model. Phase saturations were thus constant during the simulations. We applied the same body force to each phase, thus the global pressure drop was the same for both fluids. This eliminated the capillary end effects since there were no gradients in capillary pressures.

The two immiscible fluids flow until the relative permeabilities and the pressure drop have converged. When the system has converged and steady-state flow is established, the steady-state relative permeability of two fluids from average flow fluxes of the wetting and non-wetting fluids are calculated at several sections along the direction of the flow in the domain. Figure 4a shows an example of the initial distribution of the fluids in the domain and Figure5shows the experimental relative permeability curve as well as the results of LB simulation performed at a similar capillary number. According to Figure 5the numerical results are in relatively good agreement with experimental measurements.

a b

Figure 4. (a) example of initial distribution of the fluids in steady-state simulation (b) example of invasion of wetting fluid(green) in unsteady state simulation.

Figure 5.Comparison of LB modeling results and experimental relative permeability curves (steady state)(Ca=5E-6).

4.2 Unsteady state

The unsteady-state method is widely used because it is fast and qualitatively resembles the flooding process in the oil reservoir. However, it is an indirect method. Relative permeabilities are calculated, not measured. Typically, the Johnson, Bossler and Naumann (JBN) method (Johnson et al. 1959) or its variants are used to calculate relative permeabilities from the measured production data and pressure drop. This method is based on the assumptions that the flow velocity is high enough thereby making capillary end effects negligible and that the flow velocity is constant. In addition, the flow components should behave as immiscible and incompressible fluids comprising a stable displacement. Numerical simulation of the variation of relative permeabilities under unsteady- state situation is a difficult task that has not been performed before. Here, by using the developed LBM code and employing MRT technique an attempt has been made

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to model this experiment at the pore scale. To set up our unsteady-state simulations, we use periodic external boundary conditions. Both fluids could exit the model, but only the displacing fluid can enter the model. This makes the velocity field continuous during the displacement and enhances the stability of the simulations. The pressure field was controlled by a body force that was applied equally to both fluids.

6 REFRENCES

Chen H., Chen S. and Matthaeus W. H.,(1992), Recovery of the Navier–Stokes equations using a lattice-gas Boltzmann methods. Phys. Rev. A, 45: 5339–5342.

Chen S. and Doolen G. (1998). Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics,30, 329–364.

The body force is regulated to keep a constant total mean velocity and thus constant capillary number (Ramstad et al. 2010). The effluent composition and pressure drop across the model are continuously monitored. Figure 4bshow the evolution of the wetting phase into the medium and Figure6shows the experimental relative permeability curve as well as the results of LB simulation performed at similar capillary number. The LBM code predicts the trends of the variations of relative permeabilities correctly however, the discrepancies look more for unsteady-state relative permeability curves compare to those of the steady state. One of the sources of uncertainty in the current numerical results is the unfavorable effects of spurious velocities. In a SC type LBM simulation, largest spurious velocities occur near the interfacial region of the fluids (Jia et al., 2008). Therefore, high spurious velocities may affect the calculated fluxes especially for very slow or creep fluid flows. On the other hand, as mentioned before the experimental values are not measured directly from the tests. They are calculated using the JBN method. This may also contribute to the difference that is seen between the experimental and numerically-derived values.

Ghassemi A., Pak A.,(2011), Numerical Study of Factors Influencing Relative Permeabilities of Two Immiscible Fluids Flowing through Porous media using Lattice Boltzmann Method, Journal of Petroleum Science and Engineering.,77, 135-145.

Gunstensen AK, Rothman DH, Zaleski S, Zanetti G., (1991), Lattice Boltzmann model of immiscible fluids. Physical Review A, 43, 4320-4327.

He X., Zou Q., Luo L. S., Dembo M.,(1997), Analytic solutions and analysis on non-slip boundary condition for the lattice boltzmann BGK model. Stat Phys., 87:115–136.

Jia X., McLaughlin J. B., Kontomaris K., (2008), Lattice Boltzmann simulations of flows with fluid–fluid interfaces, Asia-pacific journal of chemical engineering, 3: 124–143.

Johnson E.F., Bossler D.D., Naumann V.O,(1959), Calculation of relative permeability from displacement experiments, Trans. AIME 216, 370.

Keehm Y., Mukerji T. and Nur A.,(2004), Relative Permeability Simulation using the Two-phase Lattice-Boltzmann Method., 5th Conference & Exposition on Petroleum Geophysics, Hyderabad-, India PP 696-703.

Lallemand P., Luo L.-S. (2000), Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Physical Review E 61:6546-6562.

Lenormand R., Zarcone C. and Sarr A., (1983), Mechanisms of the displacement of one fluid by another in a network of capillary ducts, J. Fluid Mech., 135, 337-353.

Li H., Pan C., Miller C.T., (2005), Pore-scale investigation of viscous coupling effects for two-phase flow in porous media, PHYSICAL REVIEW E 72, 026705.

Nourgaliev R.R., Theofanous T.G., Joseph D.D., (2003),The lattice Boltzmann equation method: the oretical interpretation, numerics and implications, International Journal of Multiphase Flow, 29: 117-169.

Pan C., Hilpert M. and Miller C. T., (2004), Lattice-Boltzmann simulation of two-phase flow in porous media, Water Res. Research 40, W01501.

Qian Y. H. d’Humie`res D., Lallemand P.,(1992),Lattice BGK models for Navier–Stokes equation. EurophysLett., 17:479–484.

Figure 6.Comparison of LB modeling results and experimental relative permeability curves (unsteady state)(Ca=5E-6).

Ramstad T., Idowu N. and Nardi C.,(2011),Relative Permeability Calculations from Two-Phase Flow Simulations Directly on Digital Images of Porous Rocks., Transp Porous Med,11-9877-8.

Ramstad T., Qren P.E., Bakke S.,(2010), Simulation of two phase flow in reservoir rocks using a lattice Boltzmann method. SPE J. 15(4), 917–927.

5 CONCLUSIONS

A newly developed Lattice Boltzmann-based numerical code has been described in this paper. This model is based on Shan & Chen (SC) formulation which is capable of simulating the simultaneous flow of two immiscible fluids at the pore scale considering all the important interacting effects such as interfacial tension and capillary. Using this code the variation of relative permeabilities of two-fluid flow under steady state and unsteady sate conditions has been simulated which is of utmost importance in petroleum reservoir engineering. MRT approach has been employed in the code to eliminate the problem of the dependency of the results to the viscosity. The obtained results indicate that LBM is a powerful method that can simulate complex problems pertaining flow in porous materials as well as solving difficult issues in petroleum geomechanics. Results obtained in this study about the variation of the relative permeabilities in the reservoir rock reveal that although the results for steady state two-fluid flow is quite promising, the modeling of unsteady flow warrants further investigation.

Schaap M. G., Porter M. L. Christensen B. and Wildenschild D., (2007), Comparison of pressure-saturation characteristics derived from computed tomography and lattice Boltzmann simulations, Water Res. Research 43,W12S06.

Shan X, Chen H. (1993), Lattice Boltzmann model for simulating flow with multiple phases and components, Physical Review E, 47, 1815-1819.

Succi S.,Foti E. and HigueraF., (1989), Europhys.Lett., 1989, 10, 433. Sukop, M.C. and D.T. Thorne, Jr., (2006), Lattice Boltzmann Modeling:

An introduction for geoscientists and engineers, Springer, Heidelberg, Berlin, New York 172 p.

Swift M.R., Orlandi E, Osborn W.R., Yeomans J.M., (1996), Lattice Boltzmann simulations of liquid-gas and binary fluid systems, Physical Review E, 54, 5041-5052.

Tsakiroglou C. D., Avraam D. G., Payatakees A. C.,(2007),Transient and steady-state relative permeabilities from two-phase flow experiments in planar pore networks., Advances in Water Resources, 30 ,1981–1992.

Valavanides M.S., Constantinides, G.N. and Payatakes, A.C.,(1998), Mechanistic Model of Steady-State Two-Phase Flow in Porous Media Based on Ganglion Dynamics, Transport in Porous Media, 30, pp.267-299

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Uniform effective stress equation for soil mechanics

Équation aux contraintes effectives uniformes pour la Mécanique des Sols

Shao L.-T., Liu G., Guo X.-X. Dalian University of Technology, Dalian, P. R. China

ABSTRACT: The internal force of soil skeleton may, according to its different effect, be seperated into two groups of different balance system, including one by external load, and the pore fluid pressure and the skeleton internal force by it. In this paper wedefine the soil skeleton stress as the soil skeleton internal force by the external load excluding the pore fluid pressure. Then taking the soil skeleton, pore water and pore air as independent analysis object, we educe the soil differential equation of equilibrium from the balance analysis of infinitesimal. By comparing with the soil differential equation of equilibrium of total stress after adding up the soildifferential equation of equilibrium of each phase of the soil, we can obtain the relationship expression of total soil stress, skeletonstress and pore fluid pressure, which is considered as the soil skeleton stress equation, equivalent to the effective stress equation byTerzaghi in the saturated condition. Therefore, the uniform soil mechanics effective stress equation is obtained, indicating the physical property of the effective stress equation is the interaction of inter-phase forces. The effective stress expression on the basis of the shearing strength equivalent or volume deformation equivalent can be expressed with the soil skeleton stress or pore fluid pressure as well.

RÉSUMÉ : Les efforts subis par le squelette solide d’un sol peuvent se diviser en deux parties en équilibres qui sont celles dues à lepression du fluide interstitiel et celle due au chargement externe. Dans cet article, dans l’analyse des efforts sur le squelette les effets de la pression interstitielle ne sont pas pris en compte dans un premier temps. Ensuite, on écrit les équations d’équilibre global sur un élément de volume infinitésimal en introduisant les effets des pressions de fluide et de gaz interstitiels. On analyse successivement l’équilibre de chaque phase et on additionne ensuite les équations ce qui permet de relier entre eux les efforts total, effective et depression de fluide. L’équation des efforts du squelette représente, dans le cas saturé, l’équation sur les contraintes effectives de Karl Terzaghi. On trouve ainsi une équation unifiant des contraintes effectives en Mécanique des Sols. On l’utilise pour prouver que la nature physique de l’équation des contraintes effectives est de représenter l’interaction entre les différentes phases. On peut aussi établir de manière équivalente une expression de la contrainte effective basée sur la contrainte de cisaillement ou la déformationvolumique.

KEYWORDS: Soil mechanics ; Effective stress equation ; Soil skeleton stress ; Saturated soil ; Unsaturated soil ; Equivalent stress

1 INTRODUCTION

The soil is the multiple-phase body of the soil skeleton and pore fluid, of which the former forms the soil structure The soil deformation and shearing strength is considered as the soil skeleton deformation and shearing strength.

The analysis shall, according to the present soil mechanics study method, be made on the soil internal force, with the whole soil including the pore fluid as the object and obtain the effective stress controlling the soil deformation and strength with the introduction of the effective stress principle. In terms of effective stress, some scholars consider it as the equivalent stress educed from the soil strength or deformation equivalent while some others regard it as the soil skeleton stress. However, the effective stress principle has no sufficient theoretical basis. Therefore, the principle has been in debate since it was put forward, with focus on the amendment and applicability of the effective stress equation for the saturated soil and whether there is the effective stress equation and its form for the unsaturated soil.

This thesis, taking the soil skeleton and pore water as independent analysis object, divided the force on the soil skeleton into two groups of balance system according to different effects. The thesis educes the internal force differential equation of equilibrium of the saturated and unsaturated soil with the interaction principle of inter-phase force, thus obtaining the soil skeleton stress equation, which indicates the

soil skeleton stress equation is the soil effective stress equation. The thesis makes further discussion on the effective stress principle, indicating the effective stress equation by Terzaghi is unnecessary to be amended for it’s accurately tenable for the saturated soil. The effective stress expression on the basis of the shearing strength equivalent or volume deformation equivalent may be expressed with the soil skeleton stress or pore fluid pressure as well. The study in the thesis may provide foundation for establishing the uniform soil mechanics theory of the saturated and unsaturated soil.

2 TWO BALANCE FORCE SYSTEMS ACTING ON SOIL SKELETON

To be definite, we consider the stress of the soil mixture as the total soil stress and the stress with the skeleton as the independent analysis object as the soil skeleton stress. In the internal force analysis on infinitesimal free body with the soil skeleton as the independent object, the acting forces shall be divided into two balance force systems, namely, the external load (excluding pore fluid pressure) and arising interaction forces between skeleton grains and the pore fluid pressure and arising interaction force between grains .

To place an isolated waterproof soil grain (as sand) in the water statically, if the pressure difference of different depth of the water is ignored, each point of the grain surface will bear the

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equivalent water pressure vertical to the surface. The (smoothed) internal force caused by the water pressure on any section of the grain is equivalent to the water pressure. Similarly, the average value of the stress of the soil skeleton grains resulting from the pore fluid pressure w on grain contact point (surface) shall be equivalent to wu , as indicated in Figure 1(a). Therefore, whatever the shape and property of the soil grain contacting surface are, to investigate the effect of the fluid pressure, each soil skeleton grain may be considered to be an isolated grain in the fluid. Furthermore, the average stress caused by the pore fluid pressure on any section of the soil skeleton grain is equal to wu , as indicated in Figure 1 (b). Thus, to take the soil skeleton as the free body, the average stress on the section caused by the pore fluid pressure is equal to the pore fluid pressure at the point, as indicated in Figure 1 (c).

u

In case of the pore fluid pressure, including pore water pressure or matrix suction acts on partial surface of the grain not on the whole surface of the grain, the skeleton grain still is in balance.

(a) (b) (c) (a) stress on grain contact surface (b) stress on the section of the soil grain (c) stress on the section of the soil skeleton Fig. 1 Stress of skeleton resulting from pore fluid pressure

The pore water pressure, pore air pressure and arising inter-grain action force of the grain system are in balance, without effect on the internal force of the internal force of the skeleton system. Meanwhile, the balance force system shall not affect the shearing strength and deformation of the skeleton system.

3 DIFFERENTIAL EQUATION OF EQUILIBRIUM AND SOIL SKELETON STRESS EQUATION OF SATURATED SOIL

The soil skeleton stress is defined to be the internal force resulting from the external forces excluding the pore fluid pressure acting on the soil skeleton of a unit area. Suppose the soil is homogeneous, to select the soil skeleton and pore water of the saturated soil as the free body of the independent analysis object for the internal force balance analysis with a group of inter-phase acting force, as indicated in Figure 2.

Fig. 2 Equilibrium analysis for solid and pore water phase

In Figure 2, n is the porosity of the soil, the pore water

pressure, wu

, , ,x z xz zx

sw ws sw, , positive stress and shearing stress

respectively, ws,x x z zf f f f the acting force and reacting force of the soil skeleton and pore water in the direction of x axis and z axis with same vale and opposite direction.

In the balance condition, the force acting on the skeleton and the pore water control its own state respectively. Therefore, the soil skeleton stress is also the effective stress to control the deformation and strength of the skeleton (or the soil body), which is the measurement of all external forces acting on the skeleton, exceeding the pore fluid pressure.

According to the internal force analysis figure, the equation of equilibrium of the soil skeleton and that of the pore water under the static balance state can by obtained respectively.

Soil skeleton: , w, sw s(1 ) 0ij j i i in u f X (1)

Pore water: w, sw w 0i i inu f X (2)

Where, is the soil skeleton stress, ,

, ij

0y

, , ,i j x y z

s sxX X s dzX , ,w w 0x yX X w wzX n . To add formula (1) to (2), then obtain the equation of

equilibrium after cancelling the terms of inter-phase acting force:

, w, sw 0ij j i iu X (3)

where, sw s w sw sw sw sat, 0,i i i x y zX X X X X X r . Taking the soil skeleton and pore water as a whole system

for the balance analysis, the differential equation of equilibrium of total soil stress in the static condition can be obtained:

t , sw 0ij j iX (4)To compare formula (3) and (4), then

t , , wij j ij j iju (5) where, is the total stress and is Kronecker symbol. tij ijThis is the saturated soil skeleton stress equation, consistent

with the traditional effective stress equation, where the soil skeleton stress is the generally accepted soil effective stress.

The soil skeleton stress equation indicates the relationship between the total stress and the skeleton stress and pore water pressure, of which the physical property is the interaction of forces between the soil skeleton and pore water. From the deduction of the equation of equilibrium, it’s unnecessary to use the effective stress equation in the balance analysis with the soil skeleton and pore water as the free body separately. In other words, it’s required to introduce the soil skeleton stress equation to get the effective stress to control the soil skeleton deformation and strength in the force analysis on the whole structure of the soil skeleton and pore water to obtain the differential equation of equilibrium. Besides, it's noticeable that the equation (5) is applicable for saturated soil or porous materials with communicating pores filled with water, whatever the contacting property of grains is.

4 DIFFERENTIAL EQUATION OF EQUILIBRIUM AND SOIL SKELETON STRESS EQUATION OF UNSATURATED SOIL

The soil skeleton stress is still defined to be the internal force resulting from the external forces excluding the pore fluid pressure acting on the soil skeleton of a unit area for the unsaturated soil. Selecting free bodies for balance analysis requires meeting the following two conditions: � the water and air in the communicating pores is immiscible; the interacting force of the pore water and pore air is ignored. For simple and easy understanding, it may be supposed that the pore air pressure acts on the whole surface of the soil skeleton, just as on the saturated soil. The pressure difference (matrix suction) of pore water and pore air acts on the surface of the occupied by the pore water, as indicated in Figure 3(a).

Figure 3(b,c) indicates the force condition of the free body of unsaturated soil infinitesimal element and soil skeleton in the direction of x axis. For the homogeneous soil, the area ratios occupied by the pore water and pore air on the unit area are

and respectively, and is the corresponding porosity of the phase of the pore water and that of the pore air.

/wn n /an n wn an

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(a) (b) (c) (a) Unsaturated soil infinitesimal body (b) Surface force of skeleton free body in direction of x axis (c) Effect of matrix suction on soil skeleton Fig. 3 Stress of unsaturated soil element and skeleton

Soil-water characteristic test shows that some content of water is always in the soil, however much the pressure (matrix suction) acting on the soil, which is considered as the residual water content, of which the corresponding saturation shall be Sr. The electro-mechanical interacting force of the pore water and the soil skeleton corresponding to the residual water content is so strong that the soil no longer shows the property of the fluid, but that of the solid or semi-solid. Therefore, in the force analysis on the unsaturated soil, the pore water corresponding to the residual water content may be considered as part of the soil skeleton. Now, the soil porosity is considered as , porosity of pore water phase and pore air phase ewn and ea respectively. The saturation without calculating the residual water content is the effective saturation, namely Se, indicated with the formula:

enn

1r

er

S SSS

(6)

Similar to saturated soil, according to the internal force analysis figure, we can obtain the equation of equilibrium of each phase, and the equation of equilibrium of the soil element without any term of inter-phase acting force.

( )( ), , ,( ) 1ij j e w i e a sfiiS u S u X + + - + 0=

(13)

Or (14) ( )( ), , ,0ij j a i a w sfii

u S u u X + - - + =To compare the equation (13) or (14) with the total stress

equation of equilibrium, then obtain: (1 )ij tij ij e w ij e aS u S u = - - - (15)

Or )( uuSu waeijaijtijij

This is the soil skeleton stress equation of unsaturated soil, or the relationship expression of the total soil stress, soil skeleton stress and pore water pressure and pore air pressure,

(16)

For saturated soil, e , then the soil skeleton stress equation for unsaturated soil will be that of the saturated soil, or the effective stress equation by Terzaghi.

1S

5 SOIL VOLUME CHANGING OR STRENGTH EQUIVALENT STRESS

Besides the soil skeleton stress, the forces on the soil skeleton also include the action of pore fluid pressure and arising internal force, which have different effect on the soil shearing strength and deformation. The latter only causes the volume deformation of the grains and pressure stress on contact points of soil grains, affecting the shearing strength of the soil. In case of fully considering the effect of the soil skeleton stress and pore fluid pressure, the shearing strength and volume changing expression for the unsaturated soil will be:

( ) tan tan

tanf t a e a w c a

c e a w

c u S u u a u

a S u u

(17)

t a e a w s

s e a w

V C u S u u CV

C S u u

au

(18) which can be written into in the further:

tan tan(1 ) (1 ) tantan tanc c

f t a e a wa ac u S u u

(19)

1 1s st a e a

V C CC u S uV C C

wu (20)

Where, and s is internal friction angle and coefficient of compressibility of the soil grains respectively;

C and the

shearing strength and coefficient of compressibility of the soil. From it, obtain the equivalent stress expression for unsaturated soil:

C

(1) Shearing strength equivalent:

' tan(1 )tanc

t a ea u S u u

a w (21)

(2) Soil volume changing equivalent:

' 1 st a e a

C u S u uC

w

(22)

Generally, the pore air pressure in the soil is not high, the contacting area of soil grains is small and the value of is close to zero. Then, the effect of the pore fluid pressure on the shearing strength and volume changing can be ignored,

t a e , indicating only the effect of the soil skeleton stress.

'a wu S u u

When the pore air pressure u , then 0a '

'

tan(1 )tan

1

ct e

st e w

a S u

C S uC

w

(23)

When the soil is fully saturated, e , the above mentioned formula (21) and (22) changes into the equivalent stress expression by Skempton.

1S

According to the principle of causing equivalent volume changing on the soil infinitesimal or the shearing strength equivalent, A. W. Skempton educed the equivalent stress expression of the saturated soil and made experimental verification.

Shearing strength equivalent:

ttan(1 )

tana u

w (24)

Soil volume changing equivalent: s

t (1 )C uC

w (25)

The equivalent stress expressions of the saturated soil and unsaturated soil are uniform.

6 EFFECT OF SOIL SKELETON STRESS EQUATION OF UNSATURATED SOIL

The above educed soil skeleton stress equation of unsaturated soil will be that of the saturated soil in the saturated condition, thus considered as the uniform soil skeleton stress equation of the saturated soil and unsaturated soil, which has two effects at least:

Firstly, to obtain the soil skeleton stress with the soil skeleton stress equation directly in the condition of learning the total stress and pore water and pore air pressure of any point in the soil;

Secondly, the skeleton stress is the soil effective stress. The soil skeleton is the supporting phase of the soil or the

structural phase of the soil, whose deformation and strength is the deformation and strength of the soil skeleton. The forces of the soil skeleton decide the strength and deformation of the soil skeleton. As above mentioned, the contribution of the pore water and pore air pressure on the soil strength and deformation can be ignored. Thus, the soil skeleton stress is the effective

1046


stress to decide the soil deformation and strength and the soil skeleton stress equation of the unsaturated soil can be considered as the effective stress equation of the unsaturated soil.

In terms of shearing strength of the unsaturated soil, Vanapalli and Fredlund gave the following shearing strength formula after the experiment and analysis:

' tanf t a e a wc u S u u ' (26)

We can find the stress expression in the square bracket of such formula is the above mentioned unsaturated soil skeleton stress, which indicates that the unsaturated soil shearing strength is controlled by the skeleton stress, as the same with that of saturated soil. The shearing strength formula of saturated soil and unsaturated soil is uniform with the concept of the soil skeleton stress.

No adequate experiment data is found on unsaturated soil volume changing.

7 DISCUSSION ON THE PRINCIPLE OF EFFECTIVE STRESS

The soil skeleton stress is the real internal force acting on the soil by the external load, or the effective stress by Terzaghi. The effective stress is not the virtual stress in such meaning. It is the real stress of the soil skeleton with definite physical meaning. The essence of the effective stress principle is that the soil skeleton stress decides the soil strength and deformation in case of ignoring the effect of the pore fluid pressure.

The effective stress principle is the most important one in the soil mechanic and the foundation of the modern soil mechanics.

Such principle, put forward by Terzaghi in 1936, states that the stress of any point at the soil section will be calculated with the total major stress 1 2 3 of such point. In case of the soil pores are filled with water under the stress of u , the total major stress consist of two parts: the first is ,the stress acting on the water and solid with the equivalent strength in various directions, which is called the neutral stress or pore water pressure; the second is the difference of the total stress

，，

u

and the neutral stress, namely, 1 1 , 2 2 ,

, which can act in the solid phase of the soil. u u

3 3Such part of the total major stress is considered as the main

effective stress. The changing neutral stress does not cause the volume changing actually. The neutral stress is not linked with the damaging soil in the stress conditions.

u

Porous materials (as sand, clay and concrete) is incompressible in the action to , just like the internal friction equal to zero. The measured results of the changing stress such as compression deformation and changing shearing resistance are only caused by the changing effective stress , and

.”

u

1 2 3 To sum up, the principle of effective stress consists of two

parts: the first is that the changing soil volume and shearing strength depends on the changing effective stress completely and the second is that the soil effective stress is equal to the difference of the total stress and pore water pressure.

The study in this thesis provides the theoretical foundation for the effective stress principle and also expresses that the effective stress principle by Terzaghi requires no further amendments and that the effective stress principle may be expanded to the unsaturated soil. Vanapalli and Fredlund made experiments and analysis, showing the soil skeleton stress (effective stress) controls the shearing strength of the unsaturated soil. No adequate experiment data is found on unsaturated soil volume changing.

8 CONCLUSIONS

(1) The effect of the pore water pressure and pore air pressure on the soil skeleton constitutes the balance force

system respectively, keeping the soil skeleton in balance. If the soil skeleton stress is defined as the soil skeleton internal force from the external forces excluding the pore fluid pressure, the soil skeleton stress is the effective stress by Terzaghi. Therefore, the effective stress is not the virtual internal force but the real internal force strength of the soil skeleton.

(2) The effective stress equation by Terzaghi is tenable, unnecessary to make any amendment for the saturated soil, in case of ignoring the effect of the pore water pressure on the soil strength and volume changing.

(3) On the basis of the differential equation of equilibrium of the unsaturated soil, we can obtain the relationship expression of the skeleton stress (effective stress) from the external forces excluding pore water and pore air pressure and the total stress, pore water and pore air pressure, which is considered as the soil skeleton stress (effective stress) equation:

= t a e a wu S u u The above formula will be the effective stress equation by

Terzaghi for the saturated soil. The experiments and engineering experience show that the effective stress decide the strength and deformation of the saturated soil. The experiment results by Vanapalli et al prove that the effective stress decides the shearing strength of the unsaturated soil.

(4) If consideration is taken on the effect of the pore water and pore air pressure, the equivalent expression of the shearing strength and soil volume changing equivalent shall be respectively:

' tan(1 )tanc

t a ea u S u u

a w

' 1 st a e a

C u S u uC

w

It will be the equivalent effective equation by Skempton in the saturated condition.

Therefore, the effective stress equation and equivalent stress equation of the unsaturated soil are the uniform effective stress equation and equivalent stress equation in the soil mechanics.

9 REFERENCES

TERZAGHI K. 1948. Theoretical soil Mechanics. London: Chapman and Hall Limited.

BISHOP A W. 1959.The principle of effective stress. Teknisk Ukeblad, 106(39),113–143.

BLIGHT G E. 1965. A study of effective stress for volume change. Moisture Equilibria and Moisture Changes in Soils Beneath Covered Areas. Sydney: Butterworths, 259–269.

SKEMPTON A W. 1961. Effective stress in soils, concrete and rocks, pore pressure and suction in soils. Conf organized by the British National Society of Int Society of soil Mech. and Foundation Eng. London: Butterworths, 4–16.

SHAO L T. 1996. Pore medium mechanics analysis method and its application in soil mechanics. Dalian: Dalian University of Technology

FRELUND D G, Rahardjo H. 1993. Soil Mechanics for Unsaturated Soils. New York: John Wiley & Sons, Inc.

Vanapalli S K. Fredlund D G. Pufahl D E. Clifton A W. 1996. Model for the prediction of shear strength with respect to soil suction. Canadian Geotechnical Journal, 33(3): 379-392

TERZAGHI K. 1936. The shearing resistance of saturated soils and the angle between the planes of shear. Proc 1st Conf Soil Mech, 54–56.

JENNINGS J E B, BURLAND J B. 1962. Limitations to the use of effective stresses in unsaturated soils. Géotechnique, 12: 125–144.

1047

Particulate Modeling of Sand Slurry Flow Retardation

Modélisation par les milieux granulaires de l’effet de retard de l'écoulement des boues résiduelles

Tomac I., Gutierrez M. Colorado School of Mines, Golden, CO, USA

ABSTRACT: The focus of this study is on the flow of dense sand slurries within a narrow channel, with a volumetric particleconcentration greater than 0.1 and a ratio of channel width and particle radius less than 10. In sand slurry flow processes in narrowchannels, clogging and velocity retardation often occur and are governed by the sand concentration and slurry flow rate. A numericalmodel developed in this study permits an improved understanding of the conditions which lead to particle clogging in sand slurry flowwithin narrow channels. The used numerical model couples the discrete element method (DEM) with computational fluid dynamics (CFD)to study this flow process. A user-defined contact model was developed to capture the non-linear collision of submergedparticles and walls. A damping effect, formulated using the theory of lubrication associated with a thin fluid layer between particles isassociated with the contact model. Lubrication is observed to enhance the formation of particle packs in the slurry flow, and decreasesthe kinetic energy of particle collisions.

RÉSUMÉ : Cette étude considère l’écoulement des boues de sable dense dans un canal étroit, avec une concentration en particulesvolumétrique supérieur à 0,1 et un rapport de largeur de canal moins de 10 rayon des particules. Dans les processus d’écoulement de boue de sable dans des chenaux étroits, le colmatage et le retard de la vitesse se produisent souvent et sont régis par la concentrationen sable et le débit de la pâte. Un modèle numérique développé dans cette étude permet une meilleure compréhension des conditionsqui conduisent à des particules le colmatage de l’écoulement de boue dans les chenaux étroits. Le modèle établi un couplage entre laméthode des éléments discrets et les méthodes numériques de la dynamique des fluides pour étudier ce processus. Un modèle de contact pouvant être défini par l'utilisateur a été mis au point pour représenter la collision non linéaire des particules immergées et les murs. Un effet d'amortissement, formulée en utilisant la théorie de lubrification associée à une couche mince de fluide entre lesparticules, est associé au modèle de contact. On observe que la lubrification d'améliore la formation des paquets de particules dans l'écoulement de suspension, et diminue l'énergie cinétique des collisions de particules.

KEYWORDS: sand, lubrication, discrete element model, computational fluid mechanics, dense phase flow

1 INTRODUCTION

Sand and other particulate material submerged in water and subjected to a fluid driven transport can be found in many geotechnical problems. For example, the stability of sand drains often fails under the increase groundwater pressure during the stabilization of excavation pit bottoms or consolidation process. Failure of dams and embankments can be caused by formation of localized fluid channels - pipes caused by seepage within the embankment body that carry out material and water (Fig. 1). Another potential application of the results from this study is the prediction of grouting effectiveness. A discrete element study is performed to better understand the sand-water slurry flow in narrow channels from a micro-mechanical point of view.

13

v d r

c

D v

Figure 1. Piping in embankment caused by seepage

The motivation for the micro-mechanical approach comes from the multi-phase theory, which explains that at higher sand concentrations, the flow is assumed to be in category of dense phase flows, in which the particle interaction forces dominate over the fluid drag forces in the slurry behavior. 2 METHODOLOGY

Discrete element method coupled with computational fluid dynamics (CFD-DEM) in two dimensions is used with the commercialy available Particle Flow Code (PFC2D) (Itasca Consulting Group, Inc. 2008a). Two-way particle fluid coupling is used to model a dense phase flow of medium coarse sand and water. Fluid motion is averaged over each fluid grid element, while particle motion is tracked individually. In the dense phase flow, particle–particle collisions dominate over the fluid drag force, because there is no sufficient time to respond to the local fluid dynamic forces before the next collision occurs. Dense flow is described with the relation (Crowe et al., 2011):

(1)

where v=fluid response time and c=particle response time, d =bulk density of the dispersed phase (sand), D=particle

diameter, vr=relative particle velocity, =fluid dynamic viscosity.

In order to account for collision forces with more accuracy, a new discrete element contact model is developed for particle – particle collisions suspended in the fluid, and it is based on the elasto-hydro-dynamic theory equation that incorporates a lubrication force at particle-particle impact (Davis et al., 1986).

2.1 Continuum fluid dynamics and discrete element model

The Discrete Element Method defines a system of particles that are represented by finite spherical or disc spaces (Cundall and Strack, 1979). The motion of each particle is solved using the explicit finite difference scheme. The calculation cycle in DEM is a time-stepping algorithm that consists of repeated applications of the law of motion to each particle, the force-displacement law to each contact, and the constant updating of

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wall positions. The PFC2D uses simple fluid coupling scheme (Patankar, 1980) for incompressible viscous or inviscid flow on a fixed rectangular grid aligned with and superimposed on the Cartesian axes for the DEM model. The fluid flow in the PFC2D

uses the numerical scheme of the Navier-Stokes’ equations. The effects of solids on fluid motion are introduced to the numerical scheme of the Navier–Stokes’s equations in terms of porosity and coupling force averaged over each element (Bouillard et al., 1989). Fluid-solids coupling time-step is 100 times larger than individual particles collision time-step. The drag force applied by all the particles in each fluid element to the fluid is defined as:

bf U (2)

where bf =drag force applied to the unit volume of fluid,

=coefficient for flow,U

=average relative velocity between the particles and the fluid, defined as:

(3)where u =average velocity of all particles in a given fluid element, v =fluid velocity. Different expressions for coefficient are given for porosities with values higher and lower than e=0.8 (Bouillard et al. 1989):

2 2

(1 ) (150(1 ) 1.75 ); 0.8fe e d U e

d e

(4)

1.7

(1 )4 ; 0.83

fd

U eC e

de

(5)

0.68724(1 0.15Re );Re 1000

Re

0.44;Re 1000

pp

pd

p

C

(6)

Re fp

U ed

(7)

where d =average diameter of the particles occurring in the element, Cd=turbulent drag coefficient defined in terms of particle Reynolds number Rep, e=porosity, f=fluid density, =fluid dynamic viscosity. A fluid force equal and opposite acts to the particles in each fluid element. The fluid drag force applied to individual discrete particles is:

(8)

where dragf

=fluid drag force on the particle and r=particle radius. The fluid-applied force acts at the particle center of mass, and the rotational moment is not applied to the particle. The resultant force that determines the individual particle motion is the sum of the averaged fluid drag and individual particle-particle or particle-wall collision forces and moments.

2.1.1 User defined lubrication contact model The particle contact model in PFC2D is built to model the elasto-hydrodynamic deformation of a solid elastic sphere that is immersed in a viscous fluid and in close motion toward another sphere or a wall. The model is based on the criteria for predicting whether two solid particles will stick or rebound subsequent to impact immersed in the fluid.

The lubrication force, F(t) act as contact force when two spheres are approaching each other (Davis et al., 1986):

(9)

where =fluid dynamic viscosity, a=particle radius, v=relative approaching velocity of two particles and x=distance between particle surfaces. Physically, the lubrication of a contact can be viewed as a thin layer of viscous fluid that acts as a cushion between two particle surfaces. It slows down the initial particles velocities and decreases the kinetic energy of the particles. If the balance of the lubrication force and the fluid approaching velocities causes adjacent particles slowing down to near zero, the

he active particle radius is represented in DEM by the apparent radius that is bigger than the particle we want to model (Fig. 2).

particles may stick next to each other and get trapped with the fluid and agglomerate.

The elastic rebound depends on the overlap of two particles if they are in contact with the real radii (rij<rc) and the lubrication damping force acts upon contact when it is activated. T

RT ( j)

rij

RT (i) RI ( j)RI (i)U u v

Fig

on are activated. Thecontact force logic can be written for PFC2D for the user defined

nd compiled in C++ as:

if rij ≥ 2rc = crit (10)

if r < 2rc = crit (11)

rc=real particle radius, =fluid ynamic viscosity, k is the spring stiffness, c=dashpot constant,

and lub=lubrication constant.

ure 2. Schematic of the apparent (RI) and real (RT) radii and the approaching distance, rij.

The apparent radius enables the activation of the contact and calling the contact force when the particles approach each other at a close distance. During the time-stepping procedure, if the particles are close enough that they overlap with their real radii, then the elastic rebound and the fricti

contact model, a

26 lub( 2 )

ij ij

ij ij c

v vF ma a

ij

where Fc=contact force, dij=overlap of the particles, rij=distance between the particles centers,d

Figure 3. Particle drop test results for the sand particle with diameter d=0.25mm using user-defined lubrication force contact model

The coefficient of restitution of this system is non-linear curve dependent on the approaching velocity. Particle-particle interactions are governed by the user-defined contact model, and for the purpose of calculating their motion effect of fluid motion in fluid-solids coupled scheme they are averaged in each fluid cell (Eqn. 3). Fig. 3 shows the result of restitution coefficient simulation in PFC2D using the developed user defined contact model for collision between sand particle and wall. Coefficient of restitution is the ratio between velocity after

c x r r

(2 )c c ij ijF ma k r r cv

343 (1

bdrag

ff re

)

26( ) a vF tx

1049


and before impact. At low approaching velocities lubrication effect prevails and significantly decreases particle rebound

at higher impact velocities lubrication effect lastic rebound so much.

wasest

nd concentrations. Fluid velocity decreases with concentration increase, following similar paths for all pressure differences.

velocity, while does not affect e

3 RESULTS

A study of 30 mesh sand flow in a narrow 2 mm and 4 mm wide and 0.5 m long channels reveals behavior of sand slurry flow obstructed with particle-particle interactions and particle-wall interactions caused by concentration and ratio of channel width and particle diameter. The study is simplified with respect to using a uniform particle size distribution and coarse discretization of the CFD-DEM fixed grid. Only two grid elements across the width of the narrow channel are able to be used in CFD scheme in order to keep the accuracy of the coupling with respect to the particle size. Boundary conditions of the flow are constricted to the non-slip conditions at the channel upper and lower boundary with zero fluid velocity, and a pressure difference P between the enter and exit profiles of the channel. Average fluid and sand velocites are measured across some fixed volume in the channel after the flow

ablished. In a narrow channel, sand velocities are obstructed by formation of particle packs and solids-wall interactions.

Velocity of sand is always smaller than velocity of surrounding fluid, indicating that fluid flows arround the sand particles or packs that move with the fluid, but at slower velocity. Pressure and sand concentration effect on sand velocity in the narrow channel is studied. Fig. 4 shows the change of sand velocity under different presssure differences and sand concentrations. Average sand velocity in the channel decreases with particle concentration increase, but does not follow the same law for higher and lower (dp<500Pa) pressures. Fig. 5 shows the change of fluid velocity under different presssure differences and sa

Fpressures

igure 4. Sand velocity dependence on concentration at different

Fig

(c

ll theobserved cases had velocities low enough for the flow to be considered laminar. (This can be seen in Figs. 6 and 7.)

ure 5. Sand velocity dependence on concentration at different pressures

Figs. 6 and 7 indicate that both sand and fluid follow the power-law dependence of velocity on sand concentration in log-log plot. The velocity slopes in log-log plot are parallel and increase with pressure decrease. However, if the sand velocity and fluid velocity ratio is observed, it seems that it generally rapidly decreases with pressure increase (Fig. 8) and then approaches steady value at very high pressures. This behavior can be related to stability of the flow and forming of particles clumps that force fluid to flow arround them. When fluid flows arround clumps, it flows in narrow channel and its velocity localy inceases because of that. Pressure increase enhances the frequency of particle collisions, whose damping causes

agglomeration. Results shown in Fig. 8 indicate, as well, that velocities ratio depends on particles concentrations, not only on channel pressure. For higher initial particles concentrations

v=0.39) solids and fluid velocity ratio is more sensitive on pressure change than for lower initial concentrations (cv=0.18).

Higher sand concentrations obstruct the increase of fluid phase flow velocity expected with pressure increase, as well as sand phase velocity increase, while lower concentrations have different impact on pressure dependent velocity. A

Figure 6. Average fluid velocity change with pressure

Figure 7. Average sand velocity change with pressure

igF

. 8) at a given shear rate are investigated here with respect of fluid and sand phase separately. Non-

ure 8. Average sand velocity and average fluid velocity ratio change with pressure

Attempts were made in the past to describe the sand slurry flow using the equivalent non-Newtonian parameters dependent on fluid and particle phase characteristics, such as fluid viscosity and sand volumetric concentration (Shah, 1993). The proposed formulation was based on the assumption that fluid and sand have the same velocities in the flow. However, here is shown that this assumption is not valid for the narrow channel and higher sand concentrations. The non-Newtonian parameters K’ and n’ that form apparent viscosity of the slurry (Eq

Newtonian fluid low is:

(8)

where =shear stress, =shear rate, K=non-Newtonian fluid consistency index, n=non-Newtonian fluid flow-behavior index,

K K 1 'n n naK

1050


K’

end changes and sand phase index rapidly increases. At this point the flow is characterized with pronounced formation of packs and clogs that affect the slurry flow into a channel.

=slurry consistency index, n’=slurry flow behavior index and a=slurry apparent viscosity.

The plots reveal some interesting trends. The non-Newtonian slurry index K’ does not depend on sand concentration while the index n’ is (Figs. 9 and 10). Index n’ appears to decrease with increase of concentration, but at cv=0.3 the tr

4 CONCLUSIONS

A discrete element study of sand flow in narrow channels with higher concentrations up to the maximum concentration is studied. Maximum sand concentration is a value at which the sand transport is not possible and flow stops. The maximum sand concentration depends on channel width and particle diameter ratio, and it has a very low sensitivity to the magnitude of fluid pressure. At high pressures, slightly higher concentration than maximum was able to be transported only for a limited length along the channel. Velocity of the sand and fluid velocity have different magnitudes, and the sand is being transported in a slurry at velocities lower than the fluid. At higher concentrations, sand forms packs and clogs, and fluid flows around them. Since the two phases, solid phase and fluid phase, do not have same average velocities, and it is not possible to simply describe slurry as a mixture of two phases with and substitute equivalent power-low (non-Newtonian) fluid parameters. Both fluid and sand average velocities increase proportionally with the pressure increase. However, if the sand velocity and fluid velocity ratio is observed, it seems the generally decrease with pressure increase. In other words, using higher fluid pressures less difference in sand and fluid flow velocity can be expected. Power-law behavior of phases can be captured to describe the flow, but since the two phases have different average velocites it is hard to average and come up with unique slurry flow characterization at this point. A more comprehensive study is needed to address this issue.

e 9. The K’ fluid index for fluid and sand phase Figur

Figure 10. The n’ fluid index for fluid and sand phase

At maximum sand concentration, the flow in the channel is completely stopped. Observation for both channel widths of 2mm and 4mm and pressures span 100-5000 Pa showed little dependence of clogging on the pressure difference in the channel. At higher pressures and higher concentrations, the initial velocities were larger for a while and then they decreased to a stabile level at which the flow continued. For the 2 mm wide channel (D/W=3.3, where D=channel width and W=particle diameter) the maximum volumetric sand concentration that was able to flow with a constant velocity was 0.14 while for the 4 mm wide channel it was 0.39. Attempts of model runs with higher concentrations showed a little bit of flowing but the flow stopped. Figs. 11, 12 and 13 show the clogged sand in a narrow fracture. At limiting value of sand concentration it can be seen that fluid flows arround the sand packs formed in

5 ACKNOWLEDGEMENTS

Financial support provided by the U.S. Department of Energy under DOE Grant No. DE-FE0002760 is gratefully acknowledged. The opinions expressed in this paper are those of the authors and not the DOE.

6 REFERENCES

Bouillard, J., R. Lyczkowski, and D. Gidaspow, 1989, Porosity distributions in a fluidized bed with an immersed obstacle: AIChE Journal, v. 35, p. 908-922.

Crowe, C. T., J. D. Schwarzkopf, M. Sommerfeld, and Y. Tsuji, 2011, Multiphase flows with droplets and particles, CRC press.

Cundall, P. A., and O. Strack, 1979, A discrete numerical model for granular assemblies: Geotechnique, v. 29, p. 47-65.

the channel. This study uses numerical model to observe dense-phase fluid and solids flow in narrow channel, and more comprehensive laboratory study is recommended for future research.

Davis, R. H., J. M. Serayssol, and E. Hinch, 1986, The elastohydrodynamic collision of two spheres: Journal of Fluid Mechanics, v. 163, p. 479-497.

Patankar, S. V., 1980, Numerical heat transfer and fluid flow, Hemisphere Pub.

Shah, S., 1993, Rheological characterization of hydraulic fracturing slurries: Old Production & Facilities, v. 8, p. 123-130.

Figure 11. Clogging of sand in 4mm wide channel at initial volumetricsand concentration cv = 0.49 with particles velocities vectors in direction opposite to the flow.

Figure 12. Unstable flow with formation of particles packs at initial cv = 0.39 in 4mm wide channel with fluid flow velocity vectors around packs.

Figure 13. Formation of particles packs at initial c = 0.28 in 2mm wide vchannel.

1051

A Coupled Analysis of Fluid-Particle Interactions in Granular Soils

Analyse couplée des interactions fluide-particules dans les sols granulaires

Zhao J., Shan T. Department of Civil & Environmental Engineering, Hong Kong University of Science & Technology, Hong Kong

ABSTRACT: Fluid-particle interaction is important to a variety of geotechnical applications. Particle-scale simulation may help to provide key microscopic information towards better understanding of the behavior of granular soils. This paper presents a coupledComputational Fluid Dynamics and Discrete Element Method (CFD-DEM) approach to simulate the fluid-particle interactions insoils. The granular particle system is modeled by solving the Newton’s equation of motion by DEM and the fluid (may comprise ofboth water and air) flow is simulated by solving the locally averaged Navier-Stokes equation with CFD. The coupling is consideredby exchanging such interaction forces as drag force, buoyancy force and virtual mass force between the DEM and CFD computations.The numerical tool has been benchmarked by two classic geomechanics problems for which analytical solutions are available, thesingle particle settling problem and the one-dimensional consolidation problem. In both cases good comparisons are observed. It has been further applied to the prediction of sand heap formation in water through hopper flow. It is found the pressure dip of verticalstress profile underneath the sand pile appears to be moderately reduced by the presence of water, as compared to the dry case.Characteristics of force chain network in the former case become less heterogeneous.

RÉSUMÉ : L'interaction fluide-particules est de première importance dans un grand nombre d'applications géotechniques. Dessimulations à l'échelle de la particule semblent être un moyen pertinent d'améliorer notre connaissance du comportementmicroscopique des matériaux granulaires. L'article présente une approche couplant Mécanique des Fluides et Modélisation Discrète(CFD-DEM) afin de simuler les interactions fluide-particules dans les sols. Le système granulaire est simulé par résolution deséquations du mouvement de Newton par Méthode des Eléments Discrets, et l'écoulement du fluide (gaz et/ou liquide) est simulé parrésolution de l'équation de Navier-Stokes moyennée localement. Le couplage CFD-DEM est réalisé par l'intermédiaire des forces d'interaction telles que trainée, poussée d'Archimède, et masse virtuelle. L'outil numérique a été validé sur deux problèmesgéotechniques classiques pour lesquels les solutions analytiques sont connues : la chute libre d'une particule unique dans un fluide, etla consolidation unidirectionnelle. Il a ensuite été appliqué à la prédiction de la formation d'un tas de sable dans l'eau aprèsécoulement en trémie. Il apparaît que le déficit de contrainte verticale au centre de la base du tas de sable est modérément réduit enprésence d'eau en comparaison avec le cas du sable sec. Il semble également que la présence d'eau homogénéise le réseau des chaînes de forces.

KEYWORDS: Fluid-particle interactions, granular media, coupled CFD-DEM modeling, sand pile, anisotropy.

1 INTRODUCTION

Granular media exist in frequent form of two-phase system with stationary or moving fluids in the pores. The interactions between the fluid phase and the granular particles may play a key role in affecting the overall behavior of the material, which may sometimes work favorably for us, such as in sand production of oil reservoir, but on other occasions may cause generate adverse consequences, such as in the case of internal/surface erosion of embankments and debris flow and slope failures. Conventional approaches considering the coupling between fluid and granular particles have been mainly based on continuum mechanics and mostly phenomenological in nature. They cannot fully account for the microscopic origin of fluid-particle interaction and their impact on the macroscopic behavior of granular media, and have difficulties in dealing with the dynamic interactions between fluid phase and particles as well. This paper presents a micromechanical approach to investigate the coupling behavior in granular materials. In particular, a coupled Computational Fluid Dynamics and Discrete Element Method (CFD-DEM) approach will be developed to consider the coupling. Some interaction forces typical in relevant geomechanics applications are considered in the coupled numerical schemes. The numerical tool will be benchmarked by some classic problems before being applied to the prediction of sandpiling in water.

2 APPROACH AND FORMULATION

The coupled CFD-DEM approach typically solves the followingsystem of equations governing the motions of both particles in the DEM system and the fluid cells in the CFD system

1

1

dd

dd

0

ci

ci

npc f gi

i ij i ij

ni

i ijj

ff f f f p

f f

f ff

mt

It

nn n p

tn

nt

n

U F F F

ω M

UU U U f g

U

(1)where the first two equations express the Newton’s law of motion which govern the translational and rotational motions of a granular particle i (Cundall and Strack, 1979), while the last two equations are the Navier-Stokes equation and the continuity equation governing the fluid flow which is locally averaged over a specific fluid cell (Anderson and Jackson, 1967). The variables involved in Eq. (1) are explained as follows: Up

i = the translational velocity of the considered particle; i = the

1052


translational angular velocities of the particle. Fcij = the contact

force acting on Particle i by Particle j or the wall(s); Mij= the torque acting on Particle i by Particle j or the wall(s); nc

i= the number of total contacts for Particle i; Ff

i = the particle–fluid interaction force acting on particle i; Fg

i = the gravitational force. mi = the mass of Particle i; Ii = the moment of inertia of particle i. Uf = the average velocity of the fluid cell. n denoting the porosity. f = the averaged fluid density. p = the fluid pressure in the cell; μ = the averaged viscosity; fp= the interaction force averaged by the cell volume the particles inside the cell exert on the fluid. g = the body force vector.

The proposed numerical CFD-DEM approach solves eq

DEM is the pro

uation system in (1) as follows. The fluid phase is discretized with a typical cell size several times of the average particle diameter. At each time step, the DEM package provides such information as the position and velocity of each individual particle. The positions of all particles are then matched with the fluid cells to calculate relevant information of each cell such as the porosity. By following the coarse-grid approximation method proposed by Tsuji et al. (1993) (see also, Zhu et al.,2007), the locally averaged Navier-Stockes equation is solved by the CFD program for the averaged velocity and pressure for each cell (the flow along individual pore pathways in the mixture will not be modeled by this method). These obtained averaged values for the velocity and pressure of a cell are then used to determine the drag force and buoyancy force acting on the particles in that cell. Iterative schemes may have to be invoked to ensure the convergence of relevant quantities such as the fluid velocity and pressure. When a converged solution is obtained, the information of fluid-particle interaction forces will be passed to the DEM for the next step calculation.

Key to the coupling between the CFD and the per consideration of particle-fluid interaction forces.

Targeting at geomechanics applications, three interaction forces are considered in this study: the drag force, the buoyancy force and the virtual mass force. The drag force adopts the expression by Di Felice (1994)

2 1

8p f p

d f pC d n

1d f F U U U U (2)

Where dp= the diameter of the considered particle; Cd= the particle-fluid drag coefficient which depends on the Reynolds

number of the particle Rep where Re f p

p fn d

p U U ;

ge

2

10g Re

p0.5 1.5 lo3.7 0.65e

. While for the buoyancy force, we employ the following avera density based expression

31b

fdF

6pg (3)

The virtual mass force iadded to a particle accelerating or decelerating in a fluid which ma

s considered to reflect the inertia

y deflect certain volume of the sounding fluid to move through. In this study the following expression of virtual mass force is employed:

2vm C V F v v (4) Consequently, th

vm f p p f

e three interaction forces add up to the totinteraction force considered in the CFD-D

method described in Zhao and Sto

al EM coupling system

f d b vm F F F F (5) In computing the interaction forces, a divided void fraction

han (2012a, 2013) is followed calculate and distribute the forces in the system more

accurately.


3.1 Stokes Particle Settling Problem

The coupled CFD-DEM approach has first been benchmarked by the classic problem of spherical particle settling in water which was treated analytically by Stokes (1844). In the numerical simulation, a sphere is released from the air to a container half filled with water. Detailed model setup and selection of model parameters of the numerical simulation can be found in Zhao and Shan (2012a). Presented in Fig. 1 is the predicted settling velocity of the particle in comparison with the analytical solution derived by Stokes (1844). In the figure, the prediction denoted by “B+D” indicates the simulation only the buoyancy force and drag force were considered (termed as CASE I in the sequel), while the curve denoted with “B+D+VM” was obtained by considering all three interaction forces (hereafter this case will be called CASE II).

As can be seen, both cases of numerical simulations provide reasonable predictions. The particle develops a peak velocity before entering the water at t = 0.065 s. Upon entry into water, it quickly decelerates to a steady terminal velocity at around t = 0.14 s before hitting the bottom of the container and bouncing back. A good accordance is observed between the numerical predictions with the analytical solution for both CASE I and CASE II. There are nonetheless subtle differences between the two cases. When the virtual mass force is considered in CASE II, the deceleration process of the particle during the settling between t = 0.065 s and t = 0.09 s is slightly quicker than in CASE I when it is not considered, which also renders the prediction in CASE II coincides more closely with the analytical solution than CASE I during this stage of settling. This may indicate that the consideration of virtual mass force may reflect the effect of pushing away fluid in front of the particle more reasonably. Meanwhile it is interesting to find the particle in CASE II hits the bottom of the container and bounces back slightly earlier than in CASE I. This is indeed not surprising since the consideration of virtual mass force in CASE II leads to changed velocity field in the fluid than in CASE I which induces slightly smaller drag force during the settling process. While the drag force is found the dominant one in all interaction forces, the overall velocity of the particle in CASE II is hence faster than in CASE I which render the particle to hit the bottom earlier.

Figure 1. Comparison of the predicted particle settling velocity with the Stokes’s analytical solution for a spherical particle settling from air to water (B+D: in consideration of buoyancy force and drag force only; B+D+VM: in consideration of all three interaction forces in Eq. (5)).

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(a)

(b) Figure 2. Comparison of the predicted and analytical solutions for 1D consolidation problem on (a) the top particle settlement (b) the dissipation of excess pore water pressure.

3.2 Terzaghi’s one dimensional consolidation problem

The numerical tool has also been benchmarked by another classic soil mechanics problem, one-dimensional consolidation in a soil layer with one-way drainage, for which Terzaghi (1943) has developed an analytical solution. In simulating this 1D consolidation problem, we consider a soil column comprised of 100 equal size spheres saturated in water. The specific model parameters can be referred to Zhao and Shan (2012a). All particles are initially placed at the centre line of the column without any overlap and are emerged in water. The gravitational force and buoyancy force are then switched on to allow the particles to settle to a hydrostatic state. Once the initial consolidation is finished, a surcharge load p0=100 Pa is then applied at the top of the column. The predicted settlement of the top particle in the column and the dissipation process of the excess water pressure are presented in Fig. 2, comparing against Terzaghi’s analytical solution. As shown in Fig. 2a, the predicted settlement of the top particle in both CASE I and CASE II compare well with the analytical solution. There are some discrepancies, however, in the predicted and analytical solutions for the excess pore pressure. The differences are apparently bigger at the initial stage of the loading. The reason lies in that the analytical solution assumes an instantaneous buildup of the excess pore pressure throughout the column once the surcharge is applied, while the CFD-DEM simulation needs time to build up the whole pore pressure field, which has been discussed in Zhao and Shan (2012a, b). It is also interesting to observe that the predictions by CASE II appear to be more consistent with the analytical solution than by CASE II. This indicates that more realistic prediction can be made by considering the virtual mass force.

3.3 Application: sandpiling in water

The two benchmarking problems presented above show that the coupled CFD-DEM tool is capable of providing reliable predictions on the fluid-particle interactions for geomechanics-relevant problem. In this subsection, it is further applied to the prediction of the behavior of sandpiling in water. The piling of granular media is commonplace in many engineering branches and industries, such as the open stockpiles in agriculture, chemical engineering and mining industries. The angle of repose and the stress distribution in a sand pile is a focus of research in the community of both engineering mechanics and physics. In particular, the pressure minimum in the vertical stress profile of the base of a sand pile has been an interesting phenomenon attracting much attention in granular mechanics. While a dominant body of existing studies on sandpiling has been focused on the case of dry granular materials, relevant research on sandpile formation in an environment of water is rather scarce. This latter case can indeed find useful applications in practice, ranging from silos to road and dam constructions, land reclamation and dredging, mine product and tailing handling. To gain better understanding on the stress transmission in granular piles submerged in water, the CFD-DEM approach developed has been employed to examine the behavior of sandpiling in water (see also Shan and Zhao, 2012; Zhao and Shan, 2013).

The basic setup the simulation is as follows. A uniform packing of 15000 sphere particles are poured from a hopper through a container filled with water to form conical sand piles on a circular receiving panel with a small round baffle at the bottom of the container. Fig. 3a demonstrates the setup and the flowing process of the particles which induces the fluid flow shown by small arrows in the figure. Fig. 3b depicts the final state of a stable sand pile formed on the receiving panel. It is found from the simulation that the repose angle of a sandpile formed in water is very close to that in the dry case.

Fig. 4 presents the pressure dip observed in a sandpile formed in water in comparison with the dry case. As compared to the dry case, the presence of water generally leads to a flattened pressure dip (or reduced relative pressure). Indicative information helpful to explain the pressure dip can be obtained from the contact force network of the sandpile, as is shown in Fig. 5 for both the dry case (upper figure) and the wet case (bottom figure). In the dry case, the strong force chains (thicker columns) show an appreciable orientation with an inward inclination angle of around 70 degrees. This indicates that the weights of the upper particles of the sandpile are transferred to the bottom along these inclined chains rather than along the vertical direction. The bottom center part of the sandpile is hence shielded from supporting the weights, which explains the appreciable pressure dip observed in the dry case. In comparison, in the wet case shown at the bottom of Fig. 5, the contact force chains are more preferably oriented to the vertical direction, and there in no effective shield formed to deflect the upper weight of the sandpile. This naturally leads to a much reduced pressure dip in this latter case.

4 CONCLUSIONS

A coupled CFD-DEM approach has been developed to simulate the interactions between fluid and particle system in granular media. The DEM has been employed to simulate the motions and interactions of particles for the granular particle system, while the CFD has been used to solve the locally averaged Navier-Stokes equations for the fluid flow. The interactions between fluids and particles are considered by exchanging

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Figure 5. Normal contact force network in the sand pile formed in the dry condition (top) and in water (bottom).

between the DEM and the CFD computations such interaction forces as the drag force, the buoyancy force and the virtual mass force. The coupled numerical tool has been benchmarked by two classic soil mechanics problems and has been further applied to the prediction of sandpiling in water. These examples demonstrate that the proposed method is capable of capturing the main feature of fluid-particle interaction from a microscopic point of view. It is robust and efficient and has the potential to be applied to a wider range of geomechanics problems where fluid-particle interactions are important.

5 ACKNOWLEDGEMENTS

The study was supported by Research Grants Council of Hong Kong (through GRF 622910).

6 REFERENCES

Anderson T.B. and Jackson R., 1967. Fluid mechanical description of fluidized beds. Equations of motion. Industrial & Engineering Chemistry Fundamentals 6, 527-539.

Cundall P.A. and Strack O. 1979. A discrete numerical model for granular assemblies. Géotechnique 29, 47-65.

Di Felice R. 1994. The voidage function for fluid–particle interaction systems. Int. J. Multiph. Flow 20, 153-159.

Figure 3. Simulation of sandpiling through hopper flow into a water tank. (a) During the hopper flow; (b) Final stable sand pile.

Shan T. and Zhao J.D. 2012. The role of water on the pressure dip in sand piles. The 23rd International Congress of Theoretical and Applied Mechanics (ICTAM2012). 19-24 August 2012, Beijing, China.

Stokes G.G. 1844. On the theories of internal friction of fluids in motion and of the equilibrium and motion of elastic solids. Trans. Cambr. Phil. Soc. 8(9), 287-319.

Terzaghi K. 1943. Theoretical soil mechanics. New York: Wiley. Tsuji Y., Kawaguchi T. and Tanaka T. 1993. Discrete particle

simulation of two-dimensional fluidized bed. Powder Technology77, 79-87.

Zhao J.D. and Shan T. 2012a. Coupled CFD-DEM simulation of fluid-particle interaction in geomechanics. Powder Technology, under review.

Zhao J.D. and Shan T. 2012b. Numerical modelling of fluid-particle interaction in granular media. Theoretical and Applied Mechanics Letters. In press.

Zhao J.D. and Shan T. 2013. Discrete modeling of fluid-particle interaction in soils. In Yang Q., Zhang J.M., Zheng H. & Yao Y.P. (eds) Constitutive Modeling of Geomaterials: Advances and New Application, Proceedings of the Second International Symposium on Constitutive Modeling of Geomaterials (15-16 Oct 2012, Beijing, China). Springer, pp. 297-301.

Figure 4. Profile of normalized vertical pressure at the base of sand piles for both the dry and the wet cases.

Zhu H.P., Zhou Z.Y., Yang R.Y. and Yu A.B. 2007. Discrete particle simulation of particulate systems: theoretical developments. Chemical Engineering Science 62, 3378-3396.

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Experimental study of resilient modulus of unsaturated soil at different temperatures

Etude expérimentale du module de résilience d’un sol non saturé à différentes températures

Zhou C., Ng C.W.W. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, HKSAR

ABSTRACT: Fatigue cracking and failure in asphalt and concrete layer of a road pavement are of great concerns to pavementdesigners and users. The deformation of this layer is related to resilient modulus (MR) of subgrade soil under cyclic traffic loads. In the field, subgrade soil is subjected to daily and seasonal variations of soil suction and temperature. Although thermo-hydro-mechanical behaviour of soil has attracted intense attention, suction and thermal effects on MR under cyclic loading-unloading have rarely been reported. In this study, three series of cyclic triaxial tests have been carried out to investigate MR of an unsaturated silt at different temperatures in a newly developed suction and temperature controlled cyclic triaxial apparatus. The new apparatus isemployed to investigate MR of the unsaturated silt at six different suctions (0, 30, 60, 100, 150 and 250 kPa) and two differenttemperatures (20 and 40 �). To enhance the accuracy of strain measurements, Hall-effect transducers are adopted to measure the local axial and radial deformations of each specimen. The influence of suction and temperature on MR are presented and discussed.

RÉSUMÉ : Fissuration et rupture par fatigue dans la couche d'asphalte et de béton de la chaussée sont des grandes préoccupations pour les concepteurs et les utilisateurs des chaussées. La déformation de cette couche est liée au module de résilience (MR) du sol de fondation sous charges cycliques du trafic. Le sol de fondation est soumis à des variations quotidiennes et saisonnières de la succion et de la température sur le terrain. Bien que le comportement thermo-hydro-mécanique du sol ait intensément attiré l'attention, les effets de la succion et de la température sur MR sous cycles de charge-décharge ont été peu étudiés. Dans cette étude, trois séries d'essais triaxiaux cycliques sont réalisés pour étudier le MR d’un limon non saturé à différentes températures dans un nouvel appareil de test triaxial cyclique à succion et température contrôlées. Le nouvel appareil est utilisé pour étudier le MR du limon non saturé à six succions différentes (0, 30, 60, 100, 150 et 250 kPa) et à ux températures différentes (20 et 40 °C). Afin d'améliorer la précision des mesures de contrainte, des transducteurs à effet Hall sont adoptées pour mesurer les déformations axiales et radiales locales dechaque échantillon. L'influence de la succion et de la température sur les MR est présentée et interprétée.

KEYWORDS: Unsaturated subgrade soil, resilient modulus, cyclic, suction, thermo-hydro-mechanical

1 INTRODUCTION

Fatigue deformation, cracking and failure in asphalt and concrete layer of any road pavement are of great concerns to pavement designers and users. Their incidence may be caused by many reasons such as increase in traffic volume, deterioration of asphalt and concrete, rutting of unbound granular materials and differential settlement of subgrade soils (Brown, 1997). According to Seed (1962), resilient modulus (MR), is defined as the ratio of repeated deviator stress to axial recoverable strain in cyclic triaxial test. This ratio is widely used as a stiffness parameter to determine soil deformation under cyclic traffic loads in pavement engineering (Brown, 1997).

In the field, unsaturated subgrade soil is subjected to daily and seasonal variations of pore water pressure (or soil suction) and temperature (Jin et al., 1994). It is generally recognized that the behaviour of unsaturated soil is governed by two stress state variables, namely net normal stress (- ua) and matric suction (ua -uw), where, ua and uw are total normal stress, pore air pressure and pore water pressure, respectively (Coleman, 1962). By controlling these two stress state variables in cyclic triaxial test, Yang et al. (2008) and Ng et al. (2012) observed that MR increases with an increase in suction but at a reducing rate. Although matric suction is very important for understanding MR of subgrade soil, it is rarely controlled or measured in resilient modulus tests. This is possibly due to some complexities and difficulties in suction control and measurement. Moreover, suction-controlled tests on unsaturated soil are generally time-consuming and so they are not very welcome by many engineers and researchers.

On the other hand, thermo-hydro-mechanical behaviour of soil has attracted intense attention because of its importance in the field of geo-environmental engineering and energy foundation. Previous experimental studies have illustrated that

temperature significantly affects unsaturated soil behaviour such as swelling pressure, collapse potential, shrinkage property, compressibility, water retention behaviour and shear strength (Romero et al., 2003; Tang et al., 2008; Uchaipichat and Khalili, 2009). One important examples of thermal effect on soil behaviour is that the yielding stress of soil specimen at elevated temperature is lower than that observed at room temperature. As far as the authors are aware, thermal effects on MR of unsaturated soil under cyclic loading-unloading have rarely been reported.

In this study, the influence of two stress-state variables (matric suction and net stress) and temperature on MR of unsaturated soil under cyclic loading-unloading was investigated in a newly developed suction and temperature controlled cyclic triaxial apparatus. Effects of the number of load applications were also studied.

2 TESTING APPARATUS AND MEASURING DEVICE

Figure 1 shows the newly developed suction and temperature controlled cyclic triaxial system. It consists of two main parts: a suction controlled triaxial apparatus and a heating system. The suction controlled cyclic triaxial apparatus for testing saturated and unsaturated soils was originally developed by Ng and Yung (2008) using the axis-translation technique. In addition to a conventional pore water pressure transducer installed at the bottom of soil specimen, a mid-plane suction probe can be mounted to negative pore water pressure ; at the mid-height of a specimen. More details are given by Ng and Menzies (2007); Ng and Yung (2008); Ng and Xu (2012) and Ng et al. (2012).

The heating system installed inside the triaxial cell consists of a thermostat, a cylindrical heater with air serving as circulating fluid, two small fans for circulating air and two thermocouples. One thermocouple provides feedback to thermostat for temperature control, while the other one is used

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to check the uniformity of temperature in the triaxial cell. Compared to other suction and temperature controlled triaxial systems reported in the literature, this new system, perhaps, is the first one which can test cyclic behaviour of soil under different suction and temperatures. In addition, this system is equipped with three pairs of Hall-effect transducers, measuring local axial and radial strains at center portion of each soil specimen. After calibration by a micrometer, the resolution and accuracy of each Hall-effect transducer is about 1 and 3 μm, respectively. For a specimen height of 152 mm adopted in this study, a displacement of 3 μm corresponds to an axial stain of about 0.004%.

Figure 1. Suction and temperature controlled cyclic triaxial apparatus.

3 SOIL TYPE AND SPECIMEN PREPARATION

The material tested is a completely decomposed tuff (CDT) sampled from Hong Kong. Measured liquid limit, plastic limit and plastic index are 38%, 25% and 13%, respectively. The sand, silt and clay contents are 25%, 71% and 4%, respectively (Ng and Yung, 2008). Following the Unified Soil Classification System, the CDT is described as silt (ML) (ASTM, 2006).

In order to obtain soil specimens with identical fabric, all specimens were prepared following the same method. Each triaxial specimen, 76 mm in diameter and 152 mm in height, is compacted at initial water content of about 16.3% and dry density of about 1760 kg/m3. In order to produce a uniform specimen, each specimen is compacted in 10 layers. After the completion of compaction, the height and diameter of the specimen are measured by a caliper (readable to 0.01 mm) and a PI tape (readable to 0.01 mm), respectively. The initial suction of the specimens after compaction is 95±2 kPa as measured by a high capacity suction probe.

4 TEST PROGRAM AND PROCEDURES

Three series of cyclic triaxial tests were performed to investigate effects of (i) net stress and matric suctionand (ii) temperature on MR of an unsaturated subgrade soil. Figure 2 shows the stress and thermal paths adopted in the three series of tests. The initial state of each specimen is donated by point A in the figure. Firstly, each specimen is isotropically compressed to a net confining stress of 30 kPa (A→E) at room temperature (20

℃). Depending on the test requriements, specimens are then

brought to different suction and temperature conditions at net confining pressure of 30 kPa. Stages of suction equalisation and temperature equalisation are necessary to ensure that the entire

specimen reaches the desired suction (ua -uw) and temperature conditions.

In Series 1 tests, three specimens W0T20, W30T20 and W60T20 were wetted by decreasing suction from 95 to 0, 30 and 60 kPa (i.e., E→B, E→C and E→D) at 20 ℃, respectively. To investigate suction effects on MR along the wetting path, MR of these three specimens were measured and compared In Series 2 tests, three specimens D100T20, D150T20 and D250T20 were dried to suctions of 100, 150 and 250 kPa (i.e., E→F, E→G and E→H) at 20 ℃, respectively. MR were measured and compared to study suction effects on MR along the drying path. In Series 3 tests, three specimens W0T40, W30T40 and W60T40 were wetted from 95 to 0, 30 and 60 kPa at 20 ℃ and then heated up from 20 to 40 ℃ at constant suctions (i.e., E→B→B’, E→C→C’ and E→D→D’). To investigate thermal effect on MR of unsaturated soil, measured MR of W0T40, W30T40 and W60T40 was studied and compared with that of W0T20, W30T20 and W60T20 in Series 1. More details of experimental program are summarised in Table 1.

Figure 2. Stress and thermal path of each soil specimen during stages of suction equalisation and temperature equalisation. Table 1. Details of the experimental program.

Series Specimen

identity

Matric suction

(kPa)

Temperature

(℃)

Equalization

time (day)

1-wetting W0T20 0 20 12

W30T20 30 20 7 W60T20 60 20 4

2-drying D100T20 100 20 4 D150T20 150 20 7 D250T20 250 20 13

3-thermal W0T40 0 40 15

W30T40 30 40 11 W60T40 60 40 9

Once a specimen had equalised at a given suction and temperature, it was subjected to cyclic loads to determine its MR. In each cyclic test, applied axial stress was varied with time following a haversine form while net confining pressure and temperature was maintained constant. For clarity, variations of axial stress during the first and last 10 cycles are shown in Figure 3. The difference between the maximum and minimum axial stresses is defined as cyclic stress qcyc. According to AASHTO (2003) standard for resilient modulus test, four levels of cyclic stress (i.e., 30, 40, 55 and 70 kPa) were considered and applied to each specimen in succession. At each level of qcyc, 100 cycles of loading-unloading at 1 Hz were applied. More details are given by Ng et al., (2012).

In each cyclic triaxial test, constant water content condition is maintained because the dissipation rate of excess pore water pressure is low compared to the rate of repeated traffic loads in the field. The pore water pressure was measured at the base and mid-height of each specimen, as shown in Figure 3. For clarity,

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only the first and last 10 cycles are shown in the figure. It should be noted that the variations of pore water pressures during the remaining 80 cycles are similar with those in these 20 cycles. It can be seen that pore water pressures measured at the base and mid-height vary with applied deviator stress in similar manner. The magnitude of variation of measured pore water pressure is about 10 kPa at the base and 5 kPa at the mid-height. Previous researchers found that pore water pressure measurement at the mid-height is more representative, since it is not affected by end restraint (Hight, 1982).

Figure 3. Applied deviator stress and measured pore water pressure during a cyclic triaxial test (modified from Ng et al., 2012). 5 INTERPRETATIONS OF EXPERIMENTAL RESULTS

5.1 Effects of number of load applications (N) on resilient modulus

To investigate the influence of number of load applications on resilient modulus, resilient modulus from the Nth cycle (MN

r ) is normalised by resilient modulus from the first cycle (M 1

r ). Figure 4 shows the relationship between MN

r /M1 r and N at cyclic

stress of 70 kPa, obtained from tests at six different suction (0, 30, 60, 100, 150 and 250 kPa) and two different temperatures (20 and 40 ℃) (see Figure 2 and Table 1). This figure clearly reveals two types of soil response at 20 ℃. At zero matric suction, M N

r /M 1 r increases continuously with increasing N

(obtained from W0T20). The total increase during the 100 cycles of loading-unloading is up to 20%. This is a consequence of progressive densification resulting from that the application of repeated cyclic stress (Dehlen, 1969; Ng et al., 2012). In this study, contractive volumetric strain of specimen W0T20 measured using Hall-effect transducers is 0.25% at the end of 100 cycles of loading-unloading. The decreasing volume and hence increasing dry density under cyclic loads results in an increase in MN

r /M1 r with increasing N. On the other hand, when

matric suction is equal to or larger than 30 kPa (s = 30, 60, 100, 150 and 250 kPa), MN

r /M1 r varies only slightly with N. One

reason is that volumetric strain under cyclic loads is much smaller when matric suction is equal to or larger than 30 kPa. For example, measured contractive volumetric strain at suction of 30 kPa is only 0.03%, much smaller than 0.25%. Given such a small volumetric strain as 0.03%, the variation of MN

r /M1 r with

N becomes insignificant. By studying the relationship between normalised MN

r /M1 r

and the number of load applications (N), it is evident that measured MR is sensitive to N values at zero suction but it is almost independent of N values at different suctions.

Considering temperature effects on MN r /M1

r ratios, it is also revealed in Figure 4 that there is about 5% increase in MN

r /M1 r

ratio when temperature increases from 20℃ to 40 ℃. This observation may be explained by thermal effects on the size of yield surface. Romero et al. (2003) reported that the yielding stress of unsaturated soil specimen at elevated temperature is lower than that observed at room temperature, with the same initial void ratio and suction. Given a smaller yielding stress at a higher temperature, it may be expected that the contractive

volumetric strain and hence the influence of N on MN r /M1

r is more significant at a higher temperature.

As also revealed in the figure, the variation of MN r /M1

r is negligible when N except for the specimen tested at zero suction. A steady resilient modulus was generally achieved within 100 loading-unloading cycles at suctions larger than zero.

5.2 Effects of suction and temperature on resilient modulus

Figure 5 shows the influence of qcyc on measured MR at different suctions (0, 30, 60, 150 and 250 kPa) and temperature (20 and 40 °C). Reported MR in the figure is the average resilient modulus of the last five cycles (i.e. N = 96-100). It can bes een from this figure that MR decrease with an increase in qcyc at all suctions except s = 0. For instance, MR decrease by about 40% when qcyc increase from 30 kPa to 70 kPa at a suction of 30 kPa and teperature of 20 °C (obtained from W30T20). The observed decrease of MR with an increase in qcyc is due to the non-linearity of soil stress-strain relationship. Previous studies have demonstrated that soil stiffness is high at small strain but it decays with an increase in strain level (Atkinson, 2000). In the resilient modulus tests, strain level increases with an increase in qcyc, hence measured MR decreases with an increase in qcyc.

Figure 4. Relationship between normalized resilient modulus and number of load applications at a cyclic stress of 70 kPa (modified from Ng et al., 2012).

Figure 5. The influence of cyclic stress on resilient modulus at different suction and temperature conditions (modified from Ng et al., 2012).

This figure also reveals that MR increases with increasing s

significantly, irrespective of whether it is along a drying or a wetting path. At cyclic stress of 30 kPa and temperature of 20 ℃, MR increases by up to one order of magnitude when s increases from 0 to 250 kPa. The beneficial effects of s on MR arise due to at least two possible reasons. Firstly, when a soil specimen becomes unsaturated, voids are partly filled with water and partly occupied by air, resulting in an air-water interface in each void. When there is an increase in matric suction, the radius of an air-water interface decreases and hence induces a larger normal inter-particle contact force (Mancuso et al., 2002; Wheeler et al., 2003; Ng and Yung, 2008). This

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normal inter-particle contact force provides a stabilizing effect on an unsaturated soil by inhibiting slippage at particle contacts and enhancing the shear resistance of the unsaturated soil (Wheeler et al., 2003). Secondly, an increase in s induces the shrinkage of soil specimen (Ng and Pang, 2000). Due to the stronger inter-normal force between particles and higher density, MR measured during cyclic loading-unloading is larger at higher suctions. Further inspection of this figure reveals that the relationship between MR and s is nonlinear along a wetting path, along which soil suction is smaller than initial suction. Given the same increase in s, the percentage of increase in MR is much larger in the lower suction range. At a cyclic stress of 30 kPa, MR doubles when s increases from 0 to 30 kPa, while only increases by 10% when s increases from 30 to 60 kPa. Along a drying path, the increase rate of MR with increasing s is almost constant. The different results observed in different suction ranges are likely related with AEV of a soil specimen. The different results observed in different suction ranges are probably because the bulk water effects dominate soil behaviour when matric suction is lower than AEV of soil specimen (here about 60 kPa) and meniscus water effects dominate soil behaviour when matric suction exceeds AEV (Ng and Yung, 2008).

Comparing average steady state values of MR measured at the last 5 cycles at zero suction but at two different temperatures shown in Figure 5, average MR measured at 20 ℃ (W0T20) is almost identical to that measured at 40 ℃ (obtained W0T40). At the four levels of cyclic stress, the maximum difference in MR at 20 ℃ and at 40 ℃ is about 7%. In the temperature ranges studied, the thermal effect may be considered to be negligible at zero suction. This negligible thermal effect at zero suction on MR seems to be in agreement with previous experimental evidence. Romero et al. (2003) and Uchaipichat and Khalili (2009) observed from their oedometer tests that soil stiffness during unloading seems to be independent of thermal conditions. To fully understand the thermo-hydro-mechanical effects on MR, further experimental and theoretical studies at different suction values under different temperature conditions are needed.

6 SUMMARY AND CONCLUSIONS

Three series of cyclic triaxial tests have been carried out to investigate MR of an unsaturated silt at different temperatures in a newly developed suction and temperature controlled cyclic triaxial apparatus.

By studying the relationship between normalised MN r /M1

r and the number of load applications (N), it is evident that measured MR is sensitive to N values at zero suction but it is almost insensitive to N at different suctions. At zero suction, MR measured at cyclic stress of 70 kPa increases with N by about 20% during 100 cycles of loading-unloading. When suction is equal and larger than 30 kPa, MR measured at the same cyclic stress is almost independent of N. For unsaturated CDT specimens tested, a steady resilient response was achieved within 100 cycles of loading-unloading.

For a given stress level the increase of M N r /M 1

r with increasing N is more significant at higher temperature at zero suction. This observation may be explained by the fact that yielding stress of soil specimen is smaller at higher temperature.

Measured MR is found to be dependent on cyclic stress level and suction value. It decreases with cyclic stress because soil stress-strain behaviour under cyclic loads is highly non-linear. On the other hand, MR increases significantly with suction. When suction increases from 0 to 250 kPa, MR increases by up to one order of magnitude. This is attributed to suction induced additional inter-particle normal force which stiffens soil specimen.

It is clear that more theoretical and experimental work are needed to understand unsaturated cyclic soil behaviour and

engineering properties under different suction and temperature conditions.

7 ACKNOWLEDGEMENTS

The research grant 2012CB719805 of 2012CB719800 provided by the Ministry of Science and Technology of the People's Republic of China through the National Basic Research Program (973 project) is gratefully acknowledged. In addition, the authors would like to thank the Research Grants Council of the Hong Kong Special Administrative Region (HKSAR) for financial support from research grant HKUST6/CRF/12R.

8 REFERENCES

AASHTO 2003. Standard method of test for determining the resilient modulus of soils and aggregate materials, AASHTOT Designation T307-99. American Association of State Highway and Transportation Officials, Washington, D.C.

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Technical Committee 105 - Kasetsart University

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