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Advances in Computed Tomography for Geomaterials

Advances in Computed Tomography for

Geomaterials

GeoX 2010

Edited by Khalid A Alshibli

Allen H Reed

Associate Editors Les Butler Joanne Fredrich

Jeffrey Nunn Karsten Thompson and Clinton Willson

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2010 The rights of Khalid A Alshibli and Allen H Reed to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Cataloging-in-Publication Data GeoX 2010 (2010 New Orleans La) Advances in computed tomography o geomaterials GeoX 2010 edited by Khalid A Alshibli p cm Papers presented March 1-3 2010 in New Orleans La sponsored by Louisiana Sate University and the Naval Research Laboratory Stennis Space Center Mississippi Includes bibliographical references and index ISBN 978-1-84821-179-7 1 Soil mechanics--Research--Congresses 2 Rock mechanics--Research--Congresses 3 Tomography--Congresses 4 Three-dimensional imaging in geology--Congresses 5 Materials--Testing--Congresses 6 Concrete--Analysis--Congresses 7 Radiography--Industrial--Congresses I Alshibli Khalid II Louisiana State University (Baton Rouge La) III Naval Research Laboratory (John C Stennis Space Center) IV Title TA710A1G475 2010 625122--dc22

2009048641 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-179-7

Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne

f r

Advances in Computed Tomography for Geomaterials

Advances in Computed Tomography for

Geomaterials

GeoX 2010

Edited by Khalid A Alshibli

Allen H Reed

Associate Editors Les Butler Joanne Fredrich

Jeffrey Nunn Karsten Thompson and Clinton Willson

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2010 The rights of Khalid A Alshibli and Allen H Reed to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Cataloging-in-Publication Data GeoX 2010 (2010 New Orleans La) Advances in computed tomography o geomaterials GeoX 2010 edited by Khalid A Alshibli p cm Papers presented March 1-3 2010 in New Orleans La sponsored by Louisiana Sate University and the Naval Research Laboratory Stennis Space Center Mississippi Includes bibliographical references and index ISBN 978-1-84821-179-7 1 Soil mechanics--Research--Congresses 2 Rock mechanics--Research--Congresses 3 Tomography--Congresses 4 Three-dimensional imaging in geology--Congresses 5 Materials--Testing--Congresses 6 Concrete--Analysis--Congresses 7 Radiography--Industrial--Congresses I Alshibli Khalid II Louisiana State University (Baton Rouge La) III Naval Research Laboratory (John C Stennis Space Center) IV Title TA710A1G475 2010 625122--dc22

2009048641 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-179-7

Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne

f r

Advances in Computed Tomography for

Geomaterials

GeoX 2010

Edited by Khalid A Alshibli

Allen H Reed

Associate Editors Les Butler Joanne Fredrich

Jeffrey Nunn Karsten Thompson and Clinton Willson

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2010 The rights of Khalid A Alshibli and Allen H Reed to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Cataloging-in-Publication Data GeoX 2010 (2010 New Orleans La) Advances in computed tomography o geomaterials GeoX 2010 edited by Khalid A Alshibli p cm Papers presented March 1-3 2010 in New Orleans La sponsored by Louisiana Sate University and the Naval Research Laboratory Stennis Space Center Mississippi Includes bibliographical references and index ISBN 978-1-84821-179-7 1 Soil mechanics--Research--Congresses 2 Rock mechanics--Research--Congresses 3 Tomography--Congresses 4 Three-dimensional imaging in geology--Congresses 5 Materials--Testing--Congresses 6 Concrete--Analysis--Congresses 7 Radiography--Industrial--Congresses I Alshibli Khalid II Louisiana State University (Baton Rouge La) III Naval Research Laboratory (John C Stennis Space Center) IV Title TA710A1G475 2010 625122--dc22

2009048641 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-179-7

Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne

f r

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2010 The rights of Khalid A Alshibli and Allen H Reed to be identified as the authors of this work have been asserted by them in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Cataloging-in-Publication Data GeoX 2010 (2010 New Orleans La) Advances in computed tomography o geomaterials GeoX 2010 edited by Khalid A Alshibli p cm Papers presented March 1-3 2010 in New Orleans La sponsored by Louisiana Sate University and the Naval Research Laboratory Stennis Space Center Mississippi Includes bibliographical references and index ISBN 978-1-84821-179-7 1 Soil mechanics--Research--Congresses 2 Rock mechanics--Research--Congresses 3 Tomography--Congresses 4 Three-dimensional imaging in geology--Congresses 5 Materials--Testing--Congresses 6 Concrete--Analysis--Congresses 7 Radiography--Industrial--Congresses I Alshibli Khalid II Louisiana State University (Baton Rouge La) III Naval Research Laboratory (John C Stennis Space Center) IV Title TA710A1G475 2010 625122--dc22

2009048641 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-179-7

Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne

f r

Organizing Committee

Prof Khalid A Alshibli Louisiana State University Co-Chair Dr Allen Reed Naval Research Laboratory Co-Chair

Prof Clinton Willson Louisiana State University Prof Karsten Thompson Louisiana State University

Dr Joanne Fredrich BP America Inc Prof Les Butler Louisiana State University

Prof Jeffrey A Nunn Louisiana State University

International Advisory Committee

Dr Susan Batiste University of Colorado at Boulder USA Dr Joseacute Baruchel Grenoble synchrotron France

Dr Dominique Bernard Bordeaux University and CNRS France Dr Pierre Beacutesuelle Laboratoire 3S Grenoble France

Dr Michel Bornert senior scientist and part time professor UR Navier France Prof Veerle Cnudde Ghent University Belgium

Prof Jacques Desrues CNRS - Laboratoire 3S Grenoble France Prof David Frost Georgia Institute of Technology USA

Dr Martin Van Geet Belgian Nuclear Research Center Belgium Dr Abraham ldquoAvramirdquo Grader Ingrainrocks Houston USA

Prof Richard Jardine Imperial College UK Prof Katsuhiko Kaneko Hokkaido University Japan

Dr Richard Ketcham University of Texas-Austin USA Dr Yoshiaki Kikuchi Port amp Airport Research Institute Japan

Prof Eric Landis University of Maine USA Prof Brent Lindquist Stony Brook University USA

Prof Eyad Masad Texas AampM University USA Prof Balasingam Muhunthan Washington State University USA

Prof Yuzo Obara Head of X-Earth Center Kumamoto University Japan Prof Jun Otani Kumamoto University Japan

Prof Fusao Oka Kyoto University Japan Dr Mark Rivers Argonne National Laboratory Illinois USA

Prof Carlos Santamarina Georgia Institute of Technology USA Prof Satoru Shibuya Kobe University Japan

Prof Tim Senden Australian National University Australia Prof Stein Sture University of Colorado at Boulder USA

Dr Manabu Takahashi National Institute of Advanced Industrial Science and Technology Japan

Prof Cino Viggiani J Fourier University - Laboratoire 3S Grenoble France Prof Linbing Wang Virginia Polytechnic and State University USA

Prof Ron Wong University of Calgary Canada

Table of Contents

Foreword K A ALSHIBLI A H REED xv

Keynote Paper Sand Deformation at the Grain Scale Quantified Through X-ray Imaging G VIGGIANI P BEacuteSUELLE S A HALL J DESRUES 1

Quantitative Description of Grain Contacts in a Locked Sand J FONSECA C OrsquoSULLIVAN M R COOP 17

3D Characterization of Particle Interaction Using Synchrotron Microtomography K A ALSHIBLI A HASAN 26

Characterization of the Evolving Grain-Scale Structure in a Sand Deforming under Triaxial Compression S A HALL N LENOIR G VIGGIANI P BEacuteSUELLE J DESRUES 34

Visualization of Strain Localization and Microstructures in Soils during Deformation Using Microfocus X-ray CT Y HIGO F OKA S KIMOTO T SANAGAWA M SAWADA T SATO Y MATSUSHIMA 43

Determination of 3D Displacement Fields between X-ray Computed Tomography Images Using 3D Cross-Correlation M RAZAVI B MUHUNTHAN 52

Characterization of Shear and Compaction Bands in Sandstone Using X-ray Tomography and 3D Digital Image Correlation E M CHARALAMPIDOU SA HALL S STANCHITS G VIGGIANI H LEWIS 59

x GeoX 2010

Deformation Characteristics of Tire Chips-Sand Mixture in Triaxial Compression Test by Using X-ray CT Scanning Y KIKUCHI T HIDAKA T SATO H HAZARIKA 67

Strain Field Measurements in Sand under Triaxial Compression Using X-ray CT Data and Digital Image Correlation Y WATANABE N LENOIR S A HALL J OTANI 76

Latest Developments in 3D Analysis of Geomaterials by Morpho+ V CNUDDE J VLASSENBROECK Y DE WITTE L BRABANT M N BOONE J DEWANCKELE L VAN HOOREBEKE P JACOBS 84

Quantifying Particle Shape in 3D E J GARBOCZI 93

3D Aggregate Evaluation Using Laser and X-ray Scanning L WANG C DRUTA Y ZHOU C HARRIS 101

Computation of Aggregate Contact Points Orientation and Segregation in Asphalt Specimens Using their X-ray CT Images M KUTAY 108

Integration of 3D Imaging and Discrete Element Modeling for Concrete Fracture Problems E N LANDIS J E BOLANDER 117

Application of Microfocus X-ray CT to Investigate the Frost-induced Damage Process in Cement-based Materials M A B PROMENTILLA T SUGIYAMA 124

Evaluation of the Efficiency of Self-healing in Concrete by Means of micro-CT K VAN TITTELBOOM D VAN LOO N DE BELIE P JACOBS 132

Quantification of Material Constitution in Concrete by X-ray CT Method T TEMMYO Y OBARA 140

Sealing Behavior of Fracture in Cementitious Material with Micro-Focus X-ray CT D FUKUDA Y NARA D MORI K KANEKO 148

Extraction of Effective Cement Paste Diffusivities from X-ray Microtomography Scans K KRABBENHOFT M R KARIM 156

Advances in Computed Tomography for Geomaterials xi

Contributions of X-ray CT to the Characterization of Natural Building Stones and their Disintegration J DEWANCKELE D VAN LOO J VLASSENBROECK M N BOONE V CNUDDE M A BOONE T DE KOCK L VAN HOOREBEKE P JACOBS 164

Characterization of Porous Media in Agent Transport Simulation LB HU C SAVIDGE D RIZZO N HAYDEN M DEWOOLKAR L MEADOR J W HAGADORN 172

Two Less-Used Applications of Petrophysical CT-Scanning R P KEHL S SIDDIQUI 180

Trends in CT-Scanning of Reservoir Rocks S SIDDIQUI M R H SARKER 189

3D Microanalysis of Geological Samples with High-Resolution Computed Tomography G ZACHER J SANTILLAN O BRUNKE T MAYER 197

Combination of Laboratory Micro-CT and Micro-XRF on Geological Objects M N BOONE J DEWANCKELE V CNUDDE G SILVERSMIT L VAN HOOREBEKE L VINCZE P JACOBS 205

Quantification of Physical Properties of the Transitional Phenomena in Rock from X-ray CT Image Data A SATO K TANAKA T SHIOTE K SASA 213

Deformation in Fractured Argillaceous Rock under Seepage Flow Using X-ray CT and Digital Image Correlation D TAKANO P BEacuteSUELLE J DESRUES S A HALL 222

Experimental Investigation of Rate Effects on Two-Phase Flow through Fractured Rocks Using X-ray Computed Tomography C H LEE Z T KARPYN 230

Keynote Paper Micro-Petrophysical Experiments Via Tomography and Simulation M KUMAR E LEBEDEVA Y MELEAN M MADADI A P SHEPPARD T K VARSLOT A M KINGSTON S J LATHAM R M SOK A SAKELLARIOU C H ARNS T J SENDEN M A KNACKSTEDT 238

xii GeoX 2010

Segmentation of Low-contrast Three-phase X-ray Computed Tomography Images of Porous Media P BHATTAD C S WILLSON K E THOMPSON 254

X-ray Imaging of Fluid Flow in Capillary Imbibition Experiments C DAVID L LOUIS B MENEacuteNDEZ A PONS J FORTIN S STANCHITS J M MENGUS 262

Evaluating the Influence of Wall-Roughness on Fracture Transmissivity with CT Scanning and Flow Simulations D CRANDALL G BROMHAL D MCINTYRE 270

In Situ Permeability Measurements inside Compaction Bands Using X-ray CT and Lattice Boltzmann Calculations N LENOIR J E ANDRADE W C SUN J W RUDNICKI 279

Evaluation of Porosity in Geomaterials Treated with Biogrout Considering Partial Volume Effect Y KOBAYASHI S KAWASAKI M KATO T MUKUNOKI K KANEKO 287

Image-Based Pore-Scale Modeling Using the Finite Element Method N LANE K E THOMPSON 295

Numerical Modeling of Complex Porous Media for Borehole Applications S RYU W ZHAO G LEU P M SINGER H J CHO Y KEEHM 304

Characterization of Soil Erosion due to Infiltration into Capping Layers in Landfill T MUKUNOKI Y KARASAKI N TANIGUCHI 312

On Pore Space Partitioning in Relation to Network Model Building for Fluid Flow Computation in Porous Media E PLOUGONVEN D BERNARD N COMBARET 320

3D and Geometric Information of the Pore Structure in Pressurized Clastic Sandstone M TAKAHASHI M KATO A CHANGWAN Y URUSHIMATSU Y MICHIGUCHI H PARK 328

Evaluation of Pressure-dependent Permeability in Rock by Means of the Tracer-aided X-ray CT D FUKAHORI K SUGAWARA 336

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

International Advisory Committee

Dr Susan Batiste University of Colorado at Boulder USA Dr Joseacute Baruchel Grenoble synchrotron France

Dr Dominique Bernard Bordeaux University and CNRS France Dr Pierre Beacutesuelle Laboratoire 3S Grenoble France

Dr Michel Bornert senior scientist and part time professor UR Navier France Prof Veerle Cnudde Ghent University Belgium

Prof Jacques Desrues CNRS - Laboratoire 3S Grenoble France Prof David Frost Georgia Institute of Technology USA

Dr Martin Van Geet Belgian Nuclear Research Center Belgium Dr Abraham ldquoAvramirdquo Grader Ingrainrocks Houston USA

Prof Richard Jardine Imperial College UK Prof Katsuhiko Kaneko Hokkaido University Japan

Dr Richard Ketcham University of Texas-Austin USA Dr Yoshiaki Kikuchi Port amp Airport Research Institute Japan

Prof Eric Landis University of Maine USA Prof Brent Lindquist Stony Brook University USA

Prof Eyad Masad Texas AampM University USA Prof Balasingam Muhunthan Washington State University USA

Prof Yuzo Obara Head of X-Earth Center Kumamoto University Japan Prof Jun Otani Kumamoto University Japan

Prof Fusao Oka Kyoto University Japan Dr Mark Rivers Argonne National Laboratory Illinois USA

Prof Carlos Santamarina Georgia Institute of Technology USA Prof Satoru Shibuya Kobe University Japan

Prof Tim Senden Australian National University Australia Prof Stein Sture University of Colorado at Boulder USA

Dr Manabu Takahashi National Institute of Advanced Industrial Science and Technology Japan

Prof Cino Viggiani J Fourier University - Laboratoire 3S Grenoble France Prof Linbing Wang Virginia Polytechnic and State University USA

Prof Ron Wong University of Calgary Canada

Table of Contents

Foreword K A ALSHIBLI A H REED xv

Keynote Paper Sand Deformation at the Grain Scale Quantified Through X-ray Imaging G VIGGIANI P BEacuteSUELLE S A HALL J DESRUES 1

Quantitative Description of Grain Contacts in a Locked Sand J FONSECA C OrsquoSULLIVAN M R COOP 17

3D Characterization of Particle Interaction Using Synchrotron Microtomography K A ALSHIBLI A HASAN 26

Characterization of the Evolving Grain-Scale Structure in a Sand Deforming under Triaxial Compression S A HALL N LENOIR G VIGGIANI P BEacuteSUELLE J DESRUES 34

Visualization of Strain Localization and Microstructures in Soils during Deformation Using Microfocus X-ray CT Y HIGO F OKA S KIMOTO T SANAGAWA M SAWADA T SATO Y MATSUSHIMA 43

Determination of 3D Displacement Fields between X-ray Computed Tomography Images Using 3D Cross-Correlation M RAZAVI B MUHUNTHAN 52

Characterization of Shear and Compaction Bands in Sandstone Using X-ray Tomography and 3D Digital Image Correlation E M CHARALAMPIDOU SA HALL S STANCHITS G VIGGIANI H LEWIS 59

x GeoX 2010

Deformation Characteristics of Tire Chips-Sand Mixture in Triaxial Compression Test by Using X-ray CT Scanning Y KIKUCHI T HIDAKA T SATO H HAZARIKA 67

Strain Field Measurements in Sand under Triaxial Compression Using X-ray CT Data and Digital Image Correlation Y WATANABE N LENOIR S A HALL J OTANI 76

Latest Developments in 3D Analysis of Geomaterials by Morpho+ V CNUDDE J VLASSENBROECK Y DE WITTE L BRABANT M N BOONE J DEWANCKELE L VAN HOOREBEKE P JACOBS 84

Quantifying Particle Shape in 3D E J GARBOCZI 93

3D Aggregate Evaluation Using Laser and X-ray Scanning L WANG C DRUTA Y ZHOU C HARRIS 101

Computation of Aggregate Contact Points Orientation and Segregation in Asphalt Specimens Using their X-ray CT Images M KUTAY 108

Integration of 3D Imaging and Discrete Element Modeling for Concrete Fracture Problems E N LANDIS J E BOLANDER 117

Application of Microfocus X-ray CT to Investigate the Frost-induced Damage Process in Cement-based Materials M A B PROMENTILLA T SUGIYAMA 124

Evaluation of the Efficiency of Self-healing in Concrete by Means of micro-CT K VAN TITTELBOOM D VAN LOO N DE BELIE P JACOBS 132

Quantification of Material Constitution in Concrete by X-ray CT Method T TEMMYO Y OBARA 140

Sealing Behavior of Fracture in Cementitious Material with Micro-Focus X-ray CT D FUKUDA Y NARA D MORI K KANEKO 148

Extraction of Effective Cement Paste Diffusivities from X-ray Microtomography Scans K KRABBENHOFT M R KARIM 156

Advances in Computed Tomography for Geomaterials xi

Contributions of X-ray CT to the Characterization of Natural Building Stones and their Disintegration J DEWANCKELE D VAN LOO J VLASSENBROECK M N BOONE V CNUDDE M A BOONE T DE KOCK L VAN HOOREBEKE P JACOBS 164

Characterization of Porous Media in Agent Transport Simulation LB HU C SAVIDGE D RIZZO N HAYDEN M DEWOOLKAR L MEADOR J W HAGADORN 172

Two Less-Used Applications of Petrophysical CT-Scanning R P KEHL S SIDDIQUI 180

Trends in CT-Scanning of Reservoir Rocks S SIDDIQUI M R H SARKER 189

3D Microanalysis of Geological Samples with High-Resolution Computed Tomography G ZACHER J SANTILLAN O BRUNKE T MAYER 197

Combination of Laboratory Micro-CT and Micro-XRF on Geological Objects M N BOONE J DEWANCKELE V CNUDDE G SILVERSMIT L VAN HOOREBEKE L VINCZE P JACOBS 205

Quantification of Physical Properties of the Transitional Phenomena in Rock from X-ray CT Image Data A SATO K TANAKA T SHIOTE K SASA 213

Deformation in Fractured Argillaceous Rock under Seepage Flow Using X-ray CT and Digital Image Correlation D TAKANO P BEacuteSUELLE J DESRUES S A HALL 222

Experimental Investigation of Rate Effects on Two-Phase Flow through Fractured Rocks Using X-ray Computed Tomography C H LEE Z T KARPYN 230

Keynote Paper Micro-Petrophysical Experiments Via Tomography and Simulation M KUMAR E LEBEDEVA Y MELEAN M MADADI A P SHEPPARD T K VARSLOT A M KINGSTON S J LATHAM R M SOK A SAKELLARIOU C H ARNS T J SENDEN M A KNACKSTEDT 238

xii GeoX 2010

Segmentation of Low-contrast Three-phase X-ray Computed Tomography Images of Porous Media P BHATTAD C S WILLSON K E THOMPSON 254

X-ray Imaging of Fluid Flow in Capillary Imbibition Experiments C DAVID L LOUIS B MENEacuteNDEZ A PONS J FORTIN S STANCHITS J M MENGUS 262

Evaluating the Influence of Wall-Roughness on Fracture Transmissivity with CT Scanning and Flow Simulations D CRANDALL G BROMHAL D MCINTYRE 270

In Situ Permeability Measurements inside Compaction Bands Using X-ray CT and Lattice Boltzmann Calculations N LENOIR J E ANDRADE W C SUN J W RUDNICKI 279

Evaluation of Porosity in Geomaterials Treated with Biogrout Considering Partial Volume Effect Y KOBAYASHI S KAWASAKI M KATO T MUKUNOKI K KANEKO 287

Image-Based Pore-Scale Modeling Using the Finite Element Method N LANE K E THOMPSON 295

Numerical Modeling of Complex Porous Media for Borehole Applications S RYU W ZHAO G LEU P M SINGER H J CHO Y KEEHM 304

Characterization of Soil Erosion due to Infiltration into Capping Layers in Landfill T MUKUNOKI Y KARASAKI N TANIGUCHI 312

On Pore Space Partitioning in Relation to Network Model Building for Fluid Flow Computation in Porous Media E PLOUGONVEN D BERNARD N COMBARET 320

3D and Geometric Information of the Pore Structure in Pressurized Clastic Sandstone M TAKAHASHI M KATO A CHANGWAN Y URUSHIMATSU Y MICHIGUCHI H PARK 328

Evaluation of Pressure-dependent Permeability in Rock by Means of the Tracer-aided X-ray CT D FUKAHORI K SUGAWARA 336

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Table of Contents

Foreword K A ALSHIBLI A H REED xv

Keynote Paper Sand Deformation at the Grain Scale Quantified Through X-ray Imaging G VIGGIANI P BEacuteSUELLE S A HALL J DESRUES 1

Quantitative Description of Grain Contacts in a Locked Sand J FONSECA C OrsquoSULLIVAN M R COOP 17

3D Characterization of Particle Interaction Using Synchrotron Microtomography K A ALSHIBLI A HASAN 26

Characterization of the Evolving Grain-Scale Structure in a Sand Deforming under Triaxial Compression S A HALL N LENOIR G VIGGIANI P BEacuteSUELLE J DESRUES 34

Visualization of Strain Localization and Microstructures in Soils during Deformation Using Microfocus X-ray CT Y HIGO F OKA S KIMOTO T SANAGAWA M SAWADA T SATO Y MATSUSHIMA 43

Determination of 3D Displacement Fields between X-ray Computed Tomography Images Using 3D Cross-Correlation M RAZAVI B MUHUNTHAN 52

Characterization of Shear and Compaction Bands in Sandstone Using X-ray Tomography and 3D Digital Image Correlation E M CHARALAMPIDOU SA HALL S STANCHITS G VIGGIANI H LEWIS 59

x GeoX 2010

Deformation Characteristics of Tire Chips-Sand Mixture in Triaxial Compression Test by Using X-ray CT Scanning Y KIKUCHI T HIDAKA T SATO H HAZARIKA 67

Strain Field Measurements in Sand under Triaxial Compression Using X-ray CT Data and Digital Image Correlation Y WATANABE N LENOIR S A HALL J OTANI 76

Latest Developments in 3D Analysis of Geomaterials by Morpho+ V CNUDDE J VLASSENBROECK Y DE WITTE L BRABANT M N BOONE J DEWANCKELE L VAN HOOREBEKE P JACOBS 84

Quantifying Particle Shape in 3D E J GARBOCZI 93

3D Aggregate Evaluation Using Laser and X-ray Scanning L WANG C DRUTA Y ZHOU C HARRIS 101

Computation of Aggregate Contact Points Orientation and Segregation in Asphalt Specimens Using their X-ray CT Images M KUTAY 108

Integration of 3D Imaging and Discrete Element Modeling for Concrete Fracture Problems E N LANDIS J E BOLANDER 117

Application of Microfocus X-ray CT to Investigate the Frost-induced Damage Process in Cement-based Materials M A B PROMENTILLA T SUGIYAMA 124

Evaluation of the Efficiency of Self-healing in Concrete by Means of micro-CT K VAN TITTELBOOM D VAN LOO N DE BELIE P JACOBS 132

Quantification of Material Constitution in Concrete by X-ray CT Method T TEMMYO Y OBARA 140

Sealing Behavior of Fracture in Cementitious Material with Micro-Focus X-ray CT D FUKUDA Y NARA D MORI K KANEKO 148

Extraction of Effective Cement Paste Diffusivities from X-ray Microtomography Scans K KRABBENHOFT M R KARIM 156

Advances in Computed Tomography for Geomaterials xi

Contributions of X-ray CT to the Characterization of Natural Building Stones and their Disintegration J DEWANCKELE D VAN LOO J VLASSENBROECK M N BOONE V CNUDDE M A BOONE T DE KOCK L VAN HOOREBEKE P JACOBS 164

Characterization of Porous Media in Agent Transport Simulation LB HU C SAVIDGE D RIZZO N HAYDEN M DEWOOLKAR L MEADOR J W HAGADORN 172

Two Less-Used Applications of Petrophysical CT-Scanning R P KEHL S SIDDIQUI 180

Trends in CT-Scanning of Reservoir Rocks S SIDDIQUI M R H SARKER 189

3D Microanalysis of Geological Samples with High-Resolution Computed Tomography G ZACHER J SANTILLAN O BRUNKE T MAYER 197

Combination of Laboratory Micro-CT and Micro-XRF on Geological Objects M N BOONE J DEWANCKELE V CNUDDE G SILVERSMIT L VAN HOOREBEKE L VINCZE P JACOBS 205

Quantification of Physical Properties of the Transitional Phenomena in Rock from X-ray CT Image Data A SATO K TANAKA T SHIOTE K SASA 213

Deformation in Fractured Argillaceous Rock under Seepage Flow Using X-ray CT and Digital Image Correlation D TAKANO P BEacuteSUELLE J DESRUES S A HALL 222

Experimental Investigation of Rate Effects on Two-Phase Flow through Fractured Rocks Using X-ray Computed Tomography C H LEE Z T KARPYN 230

Keynote Paper Micro-Petrophysical Experiments Via Tomography and Simulation M KUMAR E LEBEDEVA Y MELEAN M MADADI A P SHEPPARD T K VARSLOT A M KINGSTON S J LATHAM R M SOK A SAKELLARIOU C H ARNS T J SENDEN M A KNACKSTEDT 238

xii GeoX 2010

Segmentation of Low-contrast Three-phase X-ray Computed Tomography Images of Porous Media P BHATTAD C S WILLSON K E THOMPSON 254

X-ray Imaging of Fluid Flow in Capillary Imbibition Experiments C DAVID L LOUIS B MENEacuteNDEZ A PONS J FORTIN S STANCHITS J M MENGUS 262

Evaluating the Influence of Wall-Roughness on Fracture Transmissivity with CT Scanning and Flow Simulations D CRANDALL G BROMHAL D MCINTYRE 270

In Situ Permeability Measurements inside Compaction Bands Using X-ray CT and Lattice Boltzmann Calculations N LENOIR J E ANDRADE W C SUN J W RUDNICKI 279

Evaluation of Porosity in Geomaterials Treated with Biogrout Considering Partial Volume Effect Y KOBAYASHI S KAWASAKI M KATO T MUKUNOKI K KANEKO 287

Image-Based Pore-Scale Modeling Using the Finite Element Method N LANE K E THOMPSON 295

Numerical Modeling of Complex Porous Media for Borehole Applications S RYU W ZHAO G LEU P M SINGER H J CHO Y KEEHM 304

Characterization of Soil Erosion due to Infiltration into Capping Layers in Landfill T MUKUNOKI Y KARASAKI N TANIGUCHI 312

On Pore Space Partitioning in Relation to Network Model Building for Fluid Flow Computation in Porous Media E PLOUGONVEN D BERNARD N COMBARET 320

3D and Geometric Information of the Pore Structure in Pressurized Clastic Sandstone M TAKAHASHI M KATO A CHANGWAN Y URUSHIMATSU Y MICHIGUCHI H PARK 328

Evaluation of Pressure-dependent Permeability in Rock by Means of the Tracer-aided X-ray CT D FUKAHORI K SUGAWARA 336

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

x GeoX 2010

Deformation Characteristics of Tire Chips-Sand Mixture in Triaxial Compression Test by Using X-ray CT Scanning Y KIKUCHI T HIDAKA T SATO H HAZARIKA 67

Strain Field Measurements in Sand under Triaxial Compression Using X-ray CT Data and Digital Image Correlation Y WATANABE N LENOIR S A HALL J OTANI 76

Latest Developments in 3D Analysis of Geomaterials by Morpho+ V CNUDDE J VLASSENBROECK Y DE WITTE L BRABANT M N BOONE J DEWANCKELE L VAN HOOREBEKE P JACOBS 84

Quantifying Particle Shape in 3D E J GARBOCZI 93

3D Aggregate Evaluation Using Laser and X-ray Scanning L WANG C DRUTA Y ZHOU C HARRIS 101

Computation of Aggregate Contact Points Orientation and Segregation in Asphalt Specimens Using their X-ray CT Images M KUTAY 108

Integration of 3D Imaging and Discrete Element Modeling for Concrete Fracture Problems E N LANDIS J E BOLANDER 117

Application of Microfocus X-ray CT to Investigate the Frost-induced Damage Process in Cement-based Materials M A B PROMENTILLA T SUGIYAMA 124

Evaluation of the Efficiency of Self-healing in Concrete by Means of micro-CT K VAN TITTELBOOM D VAN LOO N DE BELIE P JACOBS 132

Quantification of Material Constitution in Concrete by X-ray CT Method T TEMMYO Y OBARA 140

Sealing Behavior of Fracture in Cementitious Material with Micro-Focus X-ray CT D FUKUDA Y NARA D MORI K KANEKO 148

Extraction of Effective Cement Paste Diffusivities from X-ray Microtomography Scans K KRABBENHOFT M R KARIM 156

Advances in Computed Tomography for Geomaterials xi

Contributions of X-ray CT to the Characterization of Natural Building Stones and their Disintegration J DEWANCKELE D VAN LOO J VLASSENBROECK M N BOONE V CNUDDE M A BOONE T DE KOCK L VAN HOOREBEKE P JACOBS 164

Characterization of Porous Media in Agent Transport Simulation LB HU C SAVIDGE D RIZZO N HAYDEN M DEWOOLKAR L MEADOR J W HAGADORN 172

Two Less-Used Applications of Petrophysical CT-Scanning R P KEHL S SIDDIQUI 180

Trends in CT-Scanning of Reservoir Rocks S SIDDIQUI M R H SARKER 189

3D Microanalysis of Geological Samples with High-Resolution Computed Tomography G ZACHER J SANTILLAN O BRUNKE T MAYER 197

Combination of Laboratory Micro-CT and Micro-XRF on Geological Objects M N BOONE J DEWANCKELE V CNUDDE G SILVERSMIT L VAN HOOREBEKE L VINCZE P JACOBS 205

Quantification of Physical Properties of the Transitional Phenomena in Rock from X-ray CT Image Data A SATO K TANAKA T SHIOTE K SASA 213

Deformation in Fractured Argillaceous Rock under Seepage Flow Using X-ray CT and Digital Image Correlation D TAKANO P BEacuteSUELLE J DESRUES S A HALL 222

Experimental Investigation of Rate Effects on Two-Phase Flow through Fractured Rocks Using X-ray Computed Tomography C H LEE Z T KARPYN 230

Keynote Paper Micro-Petrophysical Experiments Via Tomography and Simulation M KUMAR E LEBEDEVA Y MELEAN M MADADI A P SHEPPARD T K VARSLOT A M KINGSTON S J LATHAM R M SOK A SAKELLARIOU C H ARNS T J SENDEN M A KNACKSTEDT 238

xii GeoX 2010

Segmentation of Low-contrast Three-phase X-ray Computed Tomography Images of Porous Media P BHATTAD C S WILLSON K E THOMPSON 254

X-ray Imaging of Fluid Flow in Capillary Imbibition Experiments C DAVID L LOUIS B MENEacuteNDEZ A PONS J FORTIN S STANCHITS J M MENGUS 262

Evaluating the Influence of Wall-Roughness on Fracture Transmissivity with CT Scanning and Flow Simulations D CRANDALL G BROMHAL D MCINTYRE 270

In Situ Permeability Measurements inside Compaction Bands Using X-ray CT and Lattice Boltzmann Calculations N LENOIR J E ANDRADE W C SUN J W RUDNICKI 279

Evaluation of Porosity in Geomaterials Treated with Biogrout Considering Partial Volume Effect Y KOBAYASHI S KAWASAKI M KATO T MUKUNOKI K KANEKO 287

Image-Based Pore-Scale Modeling Using the Finite Element Method N LANE K E THOMPSON 295

Numerical Modeling of Complex Porous Media for Borehole Applications S RYU W ZHAO G LEU P M SINGER H J CHO Y KEEHM 304

Characterization of Soil Erosion due to Infiltration into Capping Layers in Landfill T MUKUNOKI Y KARASAKI N TANIGUCHI 312

On Pore Space Partitioning in Relation to Network Model Building for Fluid Flow Computation in Porous Media E PLOUGONVEN D BERNARD N COMBARET 320

3D and Geometric Information of the Pore Structure in Pressurized Clastic Sandstone M TAKAHASHI M KATO A CHANGWAN Y URUSHIMATSU Y MICHIGUCHI H PARK 328

Evaluation of Pressure-dependent Permeability in Rock by Means of the Tracer-aided X-ray CT D FUKAHORI K SUGAWARA 336

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials xi

Contributions of X-ray CT to the Characterization of Natural Building Stones and their Disintegration J DEWANCKELE D VAN LOO J VLASSENBROECK M N BOONE V CNUDDE M A BOONE T DE KOCK L VAN HOOREBEKE P JACOBS 164

Characterization of Porous Media in Agent Transport Simulation LB HU C SAVIDGE D RIZZO N HAYDEN M DEWOOLKAR L MEADOR J W HAGADORN 172

Two Less-Used Applications of Petrophysical CT-Scanning R P KEHL S SIDDIQUI 180

Trends in CT-Scanning of Reservoir Rocks S SIDDIQUI M R H SARKER 189

3D Microanalysis of Geological Samples with High-Resolution Computed Tomography G ZACHER J SANTILLAN O BRUNKE T MAYER 197

Combination of Laboratory Micro-CT and Micro-XRF on Geological Objects M N BOONE J DEWANCKELE V CNUDDE G SILVERSMIT L VAN HOOREBEKE L VINCZE P JACOBS 205

Quantification of Physical Properties of the Transitional Phenomena in Rock from X-ray CT Image Data A SATO K TANAKA T SHIOTE K SASA 213

Deformation in Fractured Argillaceous Rock under Seepage Flow Using X-ray CT and Digital Image Correlation D TAKANO P BEacuteSUELLE J DESRUES S A HALL 222

Experimental Investigation of Rate Effects on Two-Phase Flow through Fractured Rocks Using X-ray Computed Tomography C H LEE Z T KARPYN 230

Keynote Paper Micro-Petrophysical Experiments Via Tomography and Simulation M KUMAR E LEBEDEVA Y MELEAN M MADADI A P SHEPPARD T K VARSLOT A M KINGSTON S J LATHAM R M SOK A SAKELLARIOU C H ARNS T J SENDEN M A KNACKSTEDT 238

xii GeoX 2010

Segmentation of Low-contrast Three-phase X-ray Computed Tomography Images of Porous Media P BHATTAD C S WILLSON K E THOMPSON 254

X-ray Imaging of Fluid Flow in Capillary Imbibition Experiments C DAVID L LOUIS B MENEacuteNDEZ A PONS J FORTIN S STANCHITS J M MENGUS 262

Evaluating the Influence of Wall-Roughness on Fracture Transmissivity with CT Scanning and Flow Simulations D CRANDALL G BROMHAL D MCINTYRE 270

In Situ Permeability Measurements inside Compaction Bands Using X-ray CT and Lattice Boltzmann Calculations N LENOIR J E ANDRADE W C SUN J W RUDNICKI 279

Evaluation of Porosity in Geomaterials Treated with Biogrout Considering Partial Volume Effect Y KOBAYASHI S KAWASAKI M KATO T MUKUNOKI K KANEKO 287

Image-Based Pore-Scale Modeling Using the Finite Element Method N LANE K E THOMPSON 295

Numerical Modeling of Complex Porous Media for Borehole Applications S RYU W ZHAO G LEU P M SINGER H J CHO Y KEEHM 304

Characterization of Soil Erosion due to Infiltration into Capping Layers in Landfill T MUKUNOKI Y KARASAKI N TANIGUCHI 312

On Pore Space Partitioning in Relation to Network Model Building for Fluid Flow Computation in Porous Media E PLOUGONVEN D BERNARD N COMBARET 320

3D and Geometric Information of the Pore Structure in Pressurized Clastic Sandstone M TAKAHASHI M KATO A CHANGWAN Y URUSHIMATSU Y MICHIGUCHI H PARK 328

Evaluation of Pressure-dependent Permeability in Rock by Means of the Tracer-aided X-ray CT D FUKAHORI K SUGAWARA 336

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

xii GeoX 2010

Segmentation of Low-contrast Three-phase X-ray Computed Tomography Images of Porous Media P BHATTAD C S WILLSON K E THOMPSON 254

X-ray Imaging of Fluid Flow in Capillary Imbibition Experiments C DAVID L LOUIS B MENEacuteNDEZ A PONS J FORTIN S STANCHITS J M MENGUS 262

Evaluating the Influence of Wall-Roughness on Fracture Transmissivity with CT Scanning and Flow Simulations D CRANDALL G BROMHAL D MCINTYRE 270

In Situ Permeability Measurements inside Compaction Bands Using X-ray CT and Lattice Boltzmann Calculations N LENOIR J E ANDRADE W C SUN J W RUDNICKI 279

Evaluation of Porosity in Geomaterials Treated with Biogrout Considering Partial Volume Effect Y KOBAYASHI S KAWASAKI M KATO T MUKUNOKI K KANEKO 287

Image-Based Pore-Scale Modeling Using the Finite Element Method N LANE K E THOMPSON 295

Numerical Modeling of Complex Porous Media for Borehole Applications S RYU W ZHAO G LEU P M SINGER H J CHO Y KEEHM 304

Characterization of Soil Erosion due to Infiltration into Capping Layers in Landfill T MUKUNOKI Y KARASAKI N TANIGUCHI 312

On Pore Space Partitioning in Relation to Network Model Building for Fluid Flow Computation in Porous Media E PLOUGONVEN D BERNARD N COMBARET 320

3D and Geometric Information of the Pore Structure in Pressurized Clastic Sandstone M TAKAHASHI M KATO A CHANGWAN Y URUSHIMATSU Y MICHIGUCHI H PARK 328

Evaluation of Pressure-dependent Permeability in Rock by Means of the Tracer-aided X-ray CT D FUKAHORI K SUGAWARA 336

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials xiii

Assessment of Time-Space Evolutions of Intertidal Flat Geo-Environments Using an Industrial X-ray CT Scanner F YAMADA A TAMAKI Y OBARA 343

Keynote Paper Neutron Imaging Methods in Geoscience A KAESTNER P VONTOBEL E LEHMANN 352

Progress Towards Neutron Tomography at the US Spallation Neutron Source L G BUTLER 366

Synchrotron X-ray Micro-Tomography and Geological CO2 Sequestration P S NICO J B AJO-FRANKLIN S M BENSON A MCDOWELL D B SILIN L TOMUTSA Y WU 374

Residual CO2 Saturation Distributions in Rock Samples Measured by X-ray CT H OKABE Y TSUCHIYA C H PENTLAND S IGLAUER M J BLUNT 381

X-ray CT Imaging of Coal for Geologic Sequestration of Carbon Dioxide D H SMITH S A JIKICH 389

Comparison of X-ray CT and Discrete Element Method in the Evaluation Tunnel Face Failure B CHEVALIER D TAKANO J OTANI 397

Plugging Mechanism of Open-Ended Piles Y KIKUCHI T SATO T MIZUTANI Y MORIKAWA 406

Development of a Bending Test Apparatus for Quasi-dynamical Evaluation of a Clayey Soil Using X-ray CT Image Analysis T NAKANO T MUKUNOKI J OTANI J P GOURC 414

Author Index 423

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Foreword

Geomaterials are often the fundamental building blocks of infrastructure They are the soil sediment and rock upon which manufactured geomaterials such as asphalt composites and concrete are laid or poured Geomaterials are also a fundamental foundation of modern society providing energy through coal gas oil etc Working with these materials provides interesting complex and difficult challenges such as modification construction maintenance and repair of the building blocks as along with extraction of energy and sequestration of carbon dioxide In this book numerous techniques are presented to address issues that stem from the use and evaluation of geomaterials with computed tomography (CT) imagery

CT imagery provides a basis by which many complex structuresfeature within geomaterials can be visualized and evaluated CT sections the scanned material into small parts and then reconstructs these parts into three-dimensional images This process has seen widespread used in medical fields and has grown increasingly common in diagnosing ailments in humans At the same time CT has been applied to geomaterials which are being studied for industrial and research purposes

In this book advances in CT are presented that are built upon petroleum research conducted in the late 1980s and was first addressed by a collective international group of researchers at GeoX2003 workshop (Japan) and then again addressed by a international effort at GeoX2006 (Aussois France) GeoX2010 follows in the tradition of this great research by applying the latest tools and techniques to computed tomography in studies of geomaterials

This book is a compilation of 49 papers presented at GeoX2010 in New Orleans Louisiana USA March 1-3 2010 These papers address geomaterials from many perspectives by 1) using advanced software and numerical methods to address complex geometries efficiently and more completely 2) applying novel imaging techniques such as neutron and nanometer scale tomography as well as traditional x-ray computed tomography 3) addressing issues related to energy exploration and

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

xvi GeoX 2010

climate change 4) flow through porous media and 5) coupling computed tomography with geotechnical testing methods to address deformations and progress of failure in sand rock asphalt and concrete

Overall this compilation is a broad-based address of CT applications to geomaterials that has been made possible by the efforts of faculty members from Louisiana State University and the Naval Research Laboratory Stennis Space Center Mississippi and due to the innovation and sustained research efforts by the authors their support and their staff

Khalid A ALSHIBLI Allen H REED

All the chairs and reviewers that helped out with these papers

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Sand Deformation at the Grain Scale Quantified Through X-ray Imaging

G Viggiani mdash P Beacutesuelle mdash S A Hall mdash J Desrues

Laboratoire 3S-R University of Grenoble ndash CNRS 38041 Grenoble France cinoviggianigrenoble-inpfr pierrebesuellegrenoble-inpfr stephenhallgrenoble-inpfr jacquesdesruesgrenoble-inpfr ABSTRACT This paper presents a study of localized deformation processes in sand with grain-scale resolution Our approach combines state-of-the-art x-ray micro tomography imaging with 3D Volumetric Digital Image Correlation (3D V-DIC) techniques While x-ray imaging and DIC have in the past been applied individually to study sand deformation the combination of these two methods to study the kinematics of shear band formation at the scale of the grains is the first novel aspect of this work Moreover we have developed an original grain-scale V-DIC method that enables the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual sand grains in a specimen We present results obtained with both ldquocontinuumrdquo and ldquodiscreterdquo DIC on Hostun sand and a few preliminary results (continuum DIC only) recently obtained on ooid materials which are characterized by spheroidal layered grains

KEYWORDS strain localization granular media in-situ x-ray tomography 3D volumetric digital image correlation

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

2 GeoX 2010

1 Introduction

Shear banding the localization of deformation into thin zones of intense shearing is a phenomenon commonly observed in sand and other granular materials It has quite a practical relevance from an engineering standpoint and has been thoroughly investigated in the laboratory for decades However it should be kept in mind that in the presence of localized deformations the meaning of stress and strain variables derived from boundary measurements of loads and displacements is only nominal Therefore the most valuable experimental contributions to the understanding of shear banding are those measuring in one way or another the full field of deformation in the specimen ndash which is the only means by which test results can be appropriately interpreted (Viggiani and Hall 2008) Full-field analysis of strain localization in sand possibly started in the late 1960s in Cambridge (eg Roscoe et al 1963) and was continued over the last decades in the work of a number of groups including Grenoble see Desrues and Viggiani (2004) for a review Most of these works were performed using specifically designed plane strain devices and used a range of full-field methods the more advanced of which allowed observation of the specimen throughout loading by optical methods thereby permitting measurement of the evolving strain field In the 1960s x-ray radiography was first used to measure 2D strain fields in sand (eg Roscoe 1970) From the early 1980s x-ray tomography was used by Desrues and coworkers (see Desrues 2004 for a review) and later by Alshibli et al (2000) These studies provided valuable 3D information on localization patterning in sand and also demonstrated the potential of x-ray tomography as a quantitative tool eg for measuring the evolution of void ratio inside a shear band and its relation to critical state (Desrues et al 1996)

The recent advent of x-ray micro tomography originally with synchrotron sources and now with laboratory scanners has provided much finer spatial resolution which opens up new possibilities for understanding the mechanics of granular media (in 3D) at the scale of the grain For example Oda et al (2004) presented micro tomography images of sand grains inside a shear band showing organized structures that would not have been seen in standard x-ray tomography images (because of insufficient resolution) and that had only previously been observed in 2D thin sections (Oda and Kazama 1998)

It should be noted that the images by Oda et al (2004) were obtained post-mortem ie after testing However a full understanding of the mechanisms of (localized) deformation can only be achieved if the entire deformation process is followed throughout a test while the specimen deforms This is possible by using in-situ x-ray tomography (in-situ meaning x-ray scanning at the same time as loading) A number of such in-situ studies for triaxial tests on sand were performed using medical or industrial tomography systems (eg Desrues et al 1996 Alshibli et al 2000 Otani et al 2002) More recently Matsushima et al (2006 2007) have used synchrotron x-ray in-situ micro tomography which allowed them to identify

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials 3

individual sand grains and track their displacements throughout a triaxial test ndash note that this tracking was carried out only in 2D for a section through the specimen

The aim of the study presented in this paper was both to observe the material evolution under loading with grain-scale resolution and to image the deformation processes In recent work presented at the previous GeoX workshop (Beacutesuelle et al 2006) we applied 3D Volumetric Digital Image Correlation (V-DIC) to a sequence of x-ray tomography images taken during a triaxial test on a clay-rock specimen (see also Lenoir et al 2007) In the present paper we show results of a similar DIC-based analysis of deformation for sand specimens under triaxial compression Two different granular materials were tested Hostun sand a fine-grained angular siliceous sand with a mean grain size (D50) of about 300 microm and Caicos ooid a material characterized by spheroidal grains with D50 of about 420 microm In addition we have developed a grain-scale V-DIC method that permits the characterization of the full kinematics (ie 3D displacements and rotations) of all the individual grains in a specimen So far such a method has been applied only to Hostun sand

The structure of the paper is as follows First we describe the experimental setup for triaxial testing with concurrent x-ray micro tomography We then describe the main features of the two V-DIC methodologies (continuum and discrete) used in this study Results obtained with both methods are presented and discussed for a triaxial compression test on Hostun sand For the tests on Caicos ooid the analysis is still ongoing and the evolution of full-field incremental kinematics has been obtained only from the continuum V-DIC For both materials distinct features of localized deformation are identified and their spatial and temporal development is characterized

2 Experimental setup testing program and materials tested

The experimental results presented in this work come from two testing programs The former on Hostun sand was carried out on beamline ID15A at the European Synchrotron Radiation Facility (ESRF) in Grenoble results of such program have already been presented elsewhere (Hall et al 2009 2010a) The latter on Caicos ooid was carried out using the multi-scale x-ray CT scanner recently acquired at Laboratoire 3S-R which was designed and manufactured by RXSolutions at Annecy France (see Figure 1) In this laboratory scanner a large cabin allows for the flexible working space that is needed to perform in-situ scanning Spatial resolution can be adjusted by changing the spot size and by moving the rotation stage thus changing the distance between the x-ray source and the object to be scanned (the distance between source and detector remaining the same) Both at the ESRF and at 3S-R x-ray micro tomography allowed for high spatial resolution (in the order of a few microns) which is crucial for understanding mechanics down to the grain scale It should be noted that for a given spatial resolution using a synchrotron source

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

4 GeoX 2010

provides very fast scanning thanks to the high photon flux (12 minutes for a scan of the entire sample as opposed to about 3 hours for our laboratory x-ray scanner)

Figure 1 X-ray CT scanner at Laboratoire 3S-R Overall view of the large cabin (internal dimensions height 290 cm width 175 cm depth 135 cm) and remote computer control

(left photograph) and main components of the system (right photograph)

Figure 2 Tomography set-up for triaxial testing at the beamline ID15A at ESRF

Complete set-up on the beamline (left photograph) and zoom on the specimen inside the triaxial cell (right photograph)

The tests were conducted using a specifically built in-situ setup that could be placed in the x-ray beam allowing the specimens to be scanned under load see Figure 2 The triaxial apparatus made from Plexiglas (very transparent to x-rays) is

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials 5

practically the same as a conventional system except the much smaller size and the shape of the confining cell As opposed to conventional triaxial systems the tensile reaction force is carried by the cell walls and not by tie bars (which avoids having any obstacles to the x-rays) The axial load and hence the stress deviator are applied using a motor-driven screw actuator which also does not interfere with the tomographic x-ray scans See Lenoir (2006) and Viggiani et al (2004) for full details Note that essentially the same setup was used in the experiments at the ESRF and at 3S-R except for the fact that the loading system is above the cell in the former case and below in the latter (ie the axial piston moves upward during deviatoric loading)

Several triaxial compression tests were performed on dry specimens of Hostun sand and Caicos ooid Results from just two tests are discussed herein both performed under a confining pressure of 100 kPa Both materials were tested starting from an initially dense packing obtained by dry pluviation Deviatoric loading was strain controlled with a screw driven piston moving at 60 micrommin which corresponds to quite a low strain rate (027min for a 22 mm high specimen) As in conventional triaxial testing the specimen slenderness was equal to 2 ie the diameter of the specimen was 11 mm It should be noted that despite the small sample size (in comparison to standard triaxial tests on sands) the specimen can be considered large enough to be mechanically pertinent (ie its response can be considered representative of that of a larger mass of the material) in fact the sample comprises roughly 50000 grains for Hostun sand and 20000 for Caicos ooid These reduced dimensions were imposed by the x-ray imager width at the ESRF which was just 14 mm (the sample needed to be smaller than this to not risk passing out of the field of view although this did occur by the end of the test on Hostun sand see later) The spatial resolution (ie the voxel size) was set to 14x14x14 microm3 at the ESRF while it was slightly larger for the experiment on Caicos ooid performed in the scanner at 3S-R (where the side of the voxel was 147 microm) Such a resolution was enough to clearly identify the individual grains of both materials tested As an example Figure 3 shows two tomographic slices obtained from scanning a specimen of Hostun sand at the ESRF and at 3S-R (recall that the mean grain size of Hostun sand is around 300 microm or about 20 voxels therefore each grain contains about 5500 voxels in the tomography images) Interestingly the quality of the two slices looks very much comparable at first glance However important differences may exist between them (for example in terms of the signal-to-noise ratio which is lower in the ESRF images because unfortunately scanning parameters were not optimized) which are all but ldquominor detailsrdquo when it comes to quantitative image processing eg digital image correlation

As for the materials tested Hostun sand can be considered relatively well-known as it has been the reference sand for our laboratory as well as many other research groups in Europe It is a fine-grained angular siliceous sand coming from the Hostun quarry (Drocircme France) A uniform gradation of the natural sand (so-called

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

6 GeoX 2010

Hostun RF or S28 Hostun) was used in the testing program with 100 of the material passing a 063 mm sieve and retained on a 016 mm sieve The mean particle diameter is 030 mm and the coefficient of uniformity (Cu = D60D10) is 170

The ooid material has been provided by the Resource Sciences Laboratory of ExxonMobil Research and Engineering Co Anandale (USA) The material was collected from the Caicos platform which is the southernmost platform of the Bahamian archipelago that has significant emergent islands More precisely the material tested comes from a region there described as Ambergris Shoal which is a marine shoal depositional environment Ooids are rounded ldquocoatedrdquo (layered) sedimentary grains that are essentially pure CaCO3 X-ray diffraction indicates that they are gt96 aragonite the remainder being calcite and high magnesium calcite The grains are composed internally of concentric spherical layers that build out from a single core The gradation used in this testing program is relatively uniform with 100 of the grains passing a 060 mm sieve and retained on a 018 mm sieve The mean particle diameter is 042 mm and the coefficient of uniformity is 190

Figure 3 Horizontal slices extracted from 3D x-ray micro tomography images of a specimen

of Hostun sand obtained at the beamline ID15A at ESRF (left) and at 3S-R (right)

3 Continuum and discrete volumetric digital image correlation

Digital Image Correlation (DIC) is a method that is being used increasingly in experimental solid mechanics to measure displacement and deformation fields over surfaces (eg Pan et al 2009) or through a volume (eg Bay 2008) While the method was pioneered in the 1980s its extension to measure displacement and strain fields within solid objects eg using 3D images acquired by x-ray tomography is more recent (eg Bay et al 1999 Bornert et al 2004 Lenoir et al 2007 see Bay 2008 for a review)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials 7

Whatever dimension (2D or 3D) DIC is a mathematical tool to define the best mapping of an image into another More precisely the aim is to determine the transformation Φ that relates reference and deformed configurations of an evolving system The method is based on the fundamental assumption that at any point x the grey levels in the first image f(x) are convected into the grey levels of the second image g(x) by the transformation Φ that is g(Φ(x)) = f(x) In practice this relation is never fully satisfied because of systematic and random noise For the case of images acquired by x-ray micro tomography random noise can be high and systematic reconstruction artifacts are often present

Implementations of DIC usually involve local evaluations of the transformation Φ over cubic (for the volume case) subsets that are regularly distributed over the reference image The evaluation requires solving an optimization problem for each subset in which essentially some measure of the similarity of f(x) and g(Φ(x)) in the considered subset is maximized over a parametric set of transformations As a digital image is a discrete representation of grey levels any integral over subsets is in fact discretized into a sum over voxels Some interpolation is therefore necessary to evaluate the transformation with subvoxel accuracy see Lenoir et al (1997) for further details

It should be noted that standard implementations of the approach described above assume a continuous displacement field at least within each subset Locally the transformation is assumed to be a rigid translation or a low order (usually linear or quadratic) polynomial expansion of the actual transformation When deriving strain from the displacements of separate subsets continuity between subsets is assumed For this reason we refer to this DIC analysis as ldquocontinuum DICrdquo Such a procedure can be applied to study the deformation of a granular material such as sand as long as the spatial scale of the investigation remains large with respect to the grain size It may also be used at somewhat smaller scales (a few grains within the correlation subsets) under the condition that only small deformation increments are considered However a different DIC approach is possible which recognizes the granular character both of the images and the mechanical response and has therefore the specific objective of investigating the kinematics of individual sand grains In this work a ldquodiscrete DICrdquo procedure has been developed with the specific aspect that the regularly shaped and spaced subsets are replaced by subsets centered on each individual grain with a shape following the actual shape of the grain In practice the subsets include a grain plus a small surrounding layer a few voxels thick The reason for this layer is that possibly because of the relatively high noise level in the x-ray images and almost uniform x-ray absorption of the sand grains the grey level variation within a grain was not enough for DIC Adding a layer provided the extra information of grain shape which is characteristic of each individual grain If the grains are assumed to be rigid then the transformation of each subset is a rigid motion ie it involves a three component translation vector plus a rotation The latter is represented by a rotation axis and a positive angle of

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

8 GeoX 2010

rotation about this axis (the axis is parameterized by two polar angles a longitude with respect to the specimen axis and a latitude in the cross-sectional plane)

The practical implementation of this Discrete DIC comprises four consecutive steps First the image of the undeformed specimen is segmented in order to identify and label individual grains Then a mask is defined for each grain covering the grain plus a three-voxel wide layer around the grain Standard DIC procedures of CMV-3D are then applied to determine a first evaluation of the translation of each grain making use of sufficiently large cubic subsets centered on the grains Finally starting from these initial estimates the translation and rotation of each grain are determined using the discrete DIC algorithm See Hall et al (2010a) for further details

In the following we present results obtained using both continuum and discrete V-DIC The former uses the code TOMOWARP which is based on the work of Hall (2006) (see also Hall et al 2010b for a 2D application to a granular material) Discrete V-DIC has been integrated into CMV-3D a code developed by Bornert et al (2004) (see also Lenoir et al 2007 for further details and an application to geomechanics)

4 Selected results

41 Hostun sand

X-ray tomography scans were carried out at key moments throughout the test which are marked by (small) relaxations in the loading curve in Figure 4 The sample stress-strain response shows a roughly linear initial trend followed by a curvature to the peak stress at around 11 nominal axial strain after which the stress drops to what is probably the beginning of a plateau after which the test was stopped and the sample unloaded

The image on the right in Figure 4 shows in 3D the grain detail which is possible to obtain for Hostun sand through x-ray tomography For the sake of clarity in the following we will show only 2D slices through this volumetric data and the subsequent V-DIC results The top row of Figure 5 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 4) These slices which are roughly perpendicular to the planar band of localized strain that developed during the test show that the specimen gradually leans to one side with a rotation of the upper platen in the latter part of the test However there is no clear evidence of localized deformation in these images Porosity maps shown in the bottom row of Figure 5 were obtained from the grey-scale images based on overlapping cubic windows of side 61 voxels (854 microm)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials 9

throughout the sample volume From these porosity fields an evolving inclined zone of localized dilation can be seen

Figure 4 Stress deviator versus axial strain curve for the deviatoric loading part

of the triaxial compression test on Hostun sand (left) and 3D rendering of the sand specimen showing the grain detail (right)

Continuum V-DIC has been carried out on consecutive pairs of 3D images to

provide the incremental displacement and strain fields (the results are thus averages over the given time interval) The key DIC parameters are the distance between the calculation nodes (which also represents the reference length for subsequent strain calculation) and the correlation window size in this analysis these were respectively 20 voxels (or 280 microm) and a cube with sides of 21 voxels (or 294 microm) reduced to 11 voxels (or 154 microm) for the sub-voxel derivation Results from this analysis indicate that despite the granular nature of the material smooth and relatively continuous displacement fields are measured Figure 6 shows vertical slices through the 3D field of maximum shear strain (ε1 ndash ε3)2 (where ε1 and ε3 are the major and minor principal strains) for increments 3-4 4-5 5-6 and 6-7 These strain images clearly show the evolution of a localized band that traverses the sample diagonally from top left to bottom right It is worth noting that this is an incremental analysis therefore indicates the deformation active in each strain increment This is different from what can be seen with accumulated porosity changes shown in Figure 5 As such it is seen from these incremental maps that the localization possibly initiated in increment 4-5 and was clearly developed in 5-6 ie before the peak load Note that localization is visible in these maps before it becomes clear in the porosity images (bottom row of Figure 5) The general picture is of a localization of shear strain and dilatancy which starts as a broad zone and

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

10 GeoX 2010

then progressively thins with loading In increment 6-7 this zone has a width of about 5 mm (ie about 17 D50) It is also clear that the localized zone is not uniform showing a degree of structure

Figure 5 Vertical slices extracted from the seven 3D x-ray micro tomography images of the sand specimen acquired throughout the triaxial compression (top row) and equivalent slices through the 3D volumes of calculated porosity (bottom row) For scale remember that the initial sample diameter was 11 mm

Discrete V-DIC has been applied to provide incremental analysis of grain

kinematics Following the procedure detailed earlier for each sand grain in the specimen a set of six scalar quantities (three displacements and three rotations) describing the kinematics of the grain has been determined From these results displacement components at any position within a grain can be deduced Despite these results having been derived from a discrete analysis they indicate a relatively continuous field of displacements even in the presence of strain localization which explains why continuum V-DIC performs well see Hall et al (2010a)

However locally (ie at the scale of the grain contact) the field can be discontinuous It should kept in mind that individual grains are identified through image segmentation and it is well possible that grains that are in contact will not appear so as they have been artificially separated Therefore we cannot differentiate from such segmented images grains that are in contact from those that are not A more detailed study of grain contact evolution in space and time would require defining contacts based on the original non-segmented images (see the paper by Hall et al in this volume which gives a snapshot of work in progress in our group)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

Advances in Computed Tomography for Geomaterials 11

Figure 6 Continuum V-DIC derived incremental maximum shear strains (as defined in text) for increments 3-4 4-5 5-6 and 6-7 (previous increments showed much the same picture as 3-4) The images show vertical slices through the shear strain volume near the middle of the specimen at an equivalent position to Figure 5

Figure 7 Discrete V-DIC derived incremental grain rotations for increments 3-4 4-5 5-6 and 6-7 The magnitude of grain rotations is plotted for vertical slices through the middle of the specimen at an equivalent position to Figure 6 note that the grains are plotted in the configuration at the start of the test for all increments Grains colored grey are those for which the image correlation was not successful and those colored white are those with a rotation above a threshold value of 20deg

Figure 7 shows for increments 3-4 4-5 5-6 and 6-7 the magnitude of rotation for each grain about its rotation axis (recall this is specific for a grain) in a vertical slice corresponding to the middle of the specimen as in Figure 6 Note that the grains in Figure 7 are represented in the configuration that existed at the beginning of the test and not in their displaced positions These images indicate that grain rotations become progressively more intense into a zone that roughly corresponds to where shear strain localizes (see Figure 6)

42 Caicos ooid

As for the test on Hostun sand x-ray tomography scans were carried out at key moments throughout the test ndash marked by relaxations in the loading curve in Figure 8 The sample stress-strain response shows a well-defined peak stress at around 6

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)

12 GeoX 2010

nominal axial strain after which the stress drops to a plateau the test was stopped at about 17 axial strain and the sample unloaded

Figure 8 Stress-strain response from the triaxial compression test on Caicos ooid

The top row of Figure 9 shows a series of vertical slices through the x-ray tomography images at different stages in the test (see stress-strain curve in Figure 7) These slices are roughly perpendicular to the planar band of localized strain that developed during the test Differently from what was observed for the test on Hostun sand (see Figure 5) the shear band is clearly visible on these images ndash starting from post-peak slice 5 and possibly already in slice 4 ie immediately after the stress deviator peak However the corresponding porosity maps (shown in the bottom row of Figure 9) clearly indicate an evolving inclined zone of localized dilation in the specimen

Figure 9 Vertical slices extracted from the six 3D x-ray micro tomography images of the specimen acquired throughout the triaxial compression (top row) and equivalent slices

through the 3D volumes of calculated porosity (bottom row)