Modelling Rock Fracturing Processes: Theories, Methods, and Applications 9783030355258, 303035525X

Citation preview

Baotang Shen Ove Stephansson Mikael Rinne Editors

Modelling Rock Fracturing Processes Theories, Methods, and Applications Second Edition

Modelling Rock Fracturing Processes

Baotang Shen • Ove Stephansson • Mikael Rinne Editors

Modelling Rock Fracturing Processes Theories, Methods, and Applications Second Edition

Editors Baotang Shen CSIRO Mineral Resources Brisbane, Queensland, Australia Shandong University of Science and Technology Qingdao, China

Ove Stephansson GFZ German Research Centre for Geosciences Potsdam, Brandenburg, Germany Royal Institute of Technology (KTH) Stockholm, Sweden

Mikael Rinne School of Engineering, Department of Civil Engineering Aalto University Espoo, Finland

ISBN 978-3-030-35524-1 ISBN 978-3-030-35525-8 https://doi.org/10.1007/978-3-030-35525-8

(eBook)

© Springer Nature Switzerland AG 2014, 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to our co-editor and co-author, a close friend and mentor, one of the greatest scholars in this field, Professor Ove Stephansson, who passed away on February 19, 2020 while still tirelessly working on this book –Baotang Shen & Mikael Rinne

Foreword

The Earth’s subsurface is a major resource for human society, including minerals, energy sources, energy storage and waste disposal capacity, and sites for civil infrastructure. As we develop our Earth’s subsurface, a basic question emerges: “How can we use this resource to achieve a better and more sustainable quality of life?” To answer such a question, one soon realizes the need to understand the properties and behavior of the rock mass. In the context of rock mechanics, the question becomes: “How do rock masses at all scales behave? Can we predict this behavior for the benefit of science and engineering?” In particular, rock fracture initiation and propagation under natural and human-induced conditions are major issues in seismicity, enhanced geothermal systems, mineral extraction, and underground nuclear waste storage. Models are simplified representations of the complex world that are needed to explain phenomena (e.g., seismicity), make predictions (e.g., geothermal lifespan), and optimize design decisions (e.g., mineral extraction approaches). To be useful, models must be designed to address particular purposes. There is no general model that can adequately address all issues of the rock mass at the same time; taking models constructed for one purpose and applying them to another can be disastrous, dangerous, or unsatisfactory. Furthermore, proper models must be based on sound physical and mathematical principles, designed to account for observations in the real world through laboratory experiments and field trials, and verified in practical full-scale applications. Because of the nonuniqueness of models, comparative results from several approaches can help the design process. Such model comparison exercises have proved to be very useful to improve our understanding of rock behavior and to add confidence in model results. With this new thematic publication, the authors have contributed significantly to geomechanics and related models, focusing on fracturing mechanisms in rock, including fracture mechanics in naturally fractured rock masses and coupling of rock mechanics with fluid pressure and heat flow. The book includes three sections related to the development of proper and useful models. In the first section, chapters are devoted to basic rock fracture models, brittle material behavior theory, and vii

viii

Foreword

mathematical representation, while other chapters describe laboratory experiments and field studies which identify processes to be included in the models. The second section introduces a number of numerical models by some of the best researchers in this field, encouraging cross-fertilization among the different modeling approaches. The third and last section presents a large number of case histories, demonstrating the many uses of models. The book serves a very important purpose as we try to make useful models of a complex world, and improving our models will remain an ongoing task. Good conceptual and experience-based models help us develop mathematical models, define our design problems, and guide our projects. This book is an open-minded exploration of the utility of rock models of various types, at many scales. At the smallest scale, dislocations in the mineral crystal lattice and boundary cracks (Griffith cracks) cause stress concentrations, which lead to strength reduction and creep in rocks. Large faults that lead to earthquakes are also rock fracture issues. Large-scale models invoke the need for coupling to other processes such as heat and fluid flow, a subject extensively addressed in several chapters. For models to be useful, the scale must fit the problem, and several micromechanics approaches described in the text are used in various ways to develop behavioral models at the engineering scale. This leads to upscaled representations and coupled simulations based on advances in particulate mechanics, fractured media representation, and network flow models. This book constructively deals with rocks at various scales, highlighting a rich array of engineering and simulation issues dealing with scale. Uncertainty is implicitly discussed throughout the chapters. Sometimes, analysis involves a single well-defined fracture surface, such as a planar fault; more often, the analyses presented deal with fracturing processes in naturally fractured rock masses. In addition to the uncertain location, shape, and condition of natural fractures in nature, there is uncertainty in the initial stress and pressure fields, in the constitutive parameters, and in the monitoring data used to interpret and manage cases. Instead of trying to specify all natural fractures in the domain, an impossible task, some authors use statistical representations of the rock mass fabric. But what distributions of stiffness, frictional, and cohesive parameters should be chosen for the joints? Are these distributions best approximated by Weibull, Gaussian, log-normal, or binomial approximations? Will properties change during the project, and in what way? This book broaches uncertainty issues in various ways from examining statistical models of behavior to use of discrete element contact models of masses with “representative” fabric. Some chapters of this work explore the inter-block displacements and contact forces that dominate fractured rock mass behavior; other sections analyze stressstrain-pressure-temperature problems using both continuum and discrete element models. CO2 sequestration, geothermal energy development, LNG storage, and mine seismicity all have aspects of discrete and continuum rock behavior. To validate and calibrate models, temperature, pressure, stress, and deformation data are needed (although stress changes are challenging to monitor). Measurements in the laboratory and in the field help reduce model uncertainty and make the various analysis methods and approaches covered in this book highly useful.

Foreword

ix

This volume is a new and important building block in our rock mechanics edifice, helping to make our models more useful in the quest to build a more sustainable world. The authors have helped us in this search for more useful models of rock fracture under conditions of complexity, coupling, and uncertainty. University of Waterloo, Waterloo, Canada Lawrence Berkeley National Laboratory, Berkeley, CA, USA Uppsala University, Uppsala, Sweden

Maurice Dusseault Chin-Fu Tsang

Acknowledgments

The cooperation between the coeditors of this book and the development of rock fracture mechanics modeling methods can be traced back to 1992 when the first and second editors restarted the activities at the Division of Engineering Geology of the KTH Royal Institute of Technology in Stockholm, Sweden. Sometime later, the third coeditor, Mikael Rinne, joined the group at KTH and added to the further development. We gratefully acknowledge the encouragement and support provided by KTH during the period 1992–2001. The first applications of the fracture mechanics code (FRACOD) to rock engineering problems were done in close cooperation with the Swedish Nuclear Power Inspectorate (SKI), now Swedish Radiation Safety Authority (SSM), in Stockholm. The code was applied to stability problems of tunnels and deposition holes of the early versions of the KBS-3 system for disposal of spent nuclear fuel. We are thankful to Dr. Johan Andersson and Dr. Fritz Kautsky at SKI for providing challenging problems and stimulating discussions and to SKI for the support of the code development. The international contact network in the field of management of spent nuclear fuel led to close cooperation with Hazama Corporation in Japan. We are grateful to Hazama for the many years of supporting our research and to Dr. Kiyoshi Amemiya for initiating our collaboration to interesting nuclear waste problems in Japan. We would also like to thank Christer Svemar and Rolf Christiansson at the Swedish Nuclear Fuel and Waste Management Company (SKB), Stockholm, for stimulating discussions and project support. During the last two decades, the Finnish Funding Agency for Technology and Innovation (TEKES), now Business Finland, has supported the numerical development through funding to Fracom Oy in Finland. We gratefully acknowledge the generous support by TEKES and the high skills of our contact persons Lasse Pöyhönen and Arto Kotipelto. The support from the Finnish nuclear waste management company, Posiva Oy, in many phases of the development and applications of FRACOD is also highly appreciated. Special thanks go to Johannes Suikkanen, Dr. Juhani Vira, Kimmo Kemppainen, and Dr. Topias Siren at Posiva Oy. xi

xii

Acknowledgments

Since 2004, several modeling studies were conducted using fracture mechanics approaches for geothermal energy development in Australia, and they were supported by Geodynamics Ltd, Green Rock Energy Ltd, and the Commonwealth Scientific and Industrial Research Organisation (CSIRO). We are very grateful for the support and close involvement in these studies by Dr. Doone Wyborn of Geodynamics Ltd, Adrian Larking and Gary Meyer of Green Rock Energy Ltd., and Dr. Rob Jeffrey of CSIRO. Over many years, Dr. Nick Barton has been one of the strongest advocators of FRACOD development. With his strong support and guidance, FRACOD has continued its improvement in particular for applications to underground tunneling. We would like to express our most sincere appreciation to Dr. Nick Barton. Since 2007, three international collaboration projects were established to develop and apply advanced numerical packages that predict the effect of coupled explicit fracturing/fluid flow/thermal processes. The projects were led and supported by CSIRO, Australia, and participated by more than 15 international organizations. We are thankful to the following participants of the project and their organizations: Drs. Eui Seob Park, Yong-Bok Jung, and Hyung Mok Kim, Korea Institute of Geoscience and Mineral Resources; Drs. Kwang Yeom Kim and Li Zhuang, Korea Institute of Civil Engineering and Building Technology; Drs. Taek Kon Kim, Jin Moo Lee, Hee Suk Lee, Tae Young Ko, and Julie Kim, SK Engineering and Construction, South Korea; Drs. Manfred Wutke, Ralf Junker, and Christian Bönneman, Leibniz Institute for Applied Geophysics, Hannover, Germany; Drs. Tobias Backers and Tobias Meier, geomecon GmbH, Germany; Prof. Günter Zimmermann and Dr. Arno Zang, GFZ German Research Centre for Geoscience, Germany; Johannes Suikkanen, Posiva Oy, Finland; Dr. Topias Siren, Aalto University, Finland; Dr. Doone Wyborn, Geodynamics Ltd., Australia; Prof. Ki-Bok Min and Dr. Linmao Xie, National University of Seoul, South Korea; Prof. Simon Loew and Dr. Martin Ziegler, Federal Institute of Technology ETH, Zurich, Switzerland; Dr. Jonny Rutqvist, Lawrence Berkeley National Laboratory, USA; Prof. Tan Yunliang, Shandong University of Science and Technology; Prof. Tang Chun’An, Dalian University of Technology, China; Prof. Weiguo Liang, Taiyuan University of Technology, China; and Dr. Bai Bin, Chinese Academy of Science, Wuhan Institute of Virology, China. The book includes several chapters written by invited leading experts outside the international collaboration team. We would like to thank Prof. Heinz Konietzky, TU Bergakademie Freiberg, Germany; Drs. Branko Damjanac, Jim Hazzard, Christine Detournay, and Peter Cundall, Itasca Consulting Group, Minneapolis, USA; and Dr. Jeoung Seok Yoon, DynaFrax UG, Germany, for their tremendous contributions to the book. The contents of the book have been reviewed and commented by two world leading scientists in this field, Prof. Chin-Fu Tsang and Prof. Maurice Dusseault. Their endorsements to the book are invaluable for us, and we would like to express our sincere thanks to them. The later numerical development and the writing of this book could not have been possible without strong support from CSIRO Mineral Resources, Australia. We

Acknowledgments

xiii

would like to thank Dr. Hua Guo for his strong support and scientific contribution to the international collaboration projects. We are very grateful to Dr. Binzhong Zhou for the development of the pre- and postprocessors to FRACOD and to Dr. Jingyu Shi for the development of the three-dimensional version FRACOD3D. The second coeditor is thankful to Director Reinhard Hüttle, Head of Section 2.6, and Fabrice Cotton of GFZ German Research Centre for Geosciences, Potsdam, Germany, for providing a stimulating environment and for supporting this work. The third coeditor thanks Aalto University in Finland, former Helsinki University of Technology. Special thanks are addressed to Prof. Pekka Särkkä and the personnel at Aalto. Rock testing and verification of the new features in the code has been an essential part in the code development. Finally, we acknowledge the very high-quality support and collaboration by Petra D. Van Steenbergen, Hermine Vloemans, Tiruptirekha Das Mahapatra, Margaret Deignan, and Solomon George, Springer Nature in publishing matters.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baotang Shen, Ove Stephansson, and Mikael Rinne

Part I

1

Theoretical Background

2

Introduction to the Theories of Rock Fracturing . . . . . . . . . . . . . . . Mikael Rinne, Ove Stephansson, Baotang Shen, and Heinz Konietzky

7

3

Laboratory Studies of 2D and 3D Rock Fracture Propagation . . . . Baotang Shen, Xizhen Sun, and Baoliang Zhang

25

4

Laboratory Investigations on the Hydraulic Fracturing of Granite Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Zhuang, Sunggyu Jung, Melvin Diaz, and Kwang Yeom Kim

5

Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and Permeability in Rock Mass at 410 m Depth in Äspö Hard Rock Laboratory, Sweden . . . . . . . . . . . . . . . . . . . . . . . . . . . Arno Zang, Ove Stephansson, and Günter Zimmermann

Part II

61

89

Numerical Methods

6

Modelling Rock Fracturing Processes with FRACOD . . . . . . . . . . . 105 Baotang Shen, Ove Stephansson, and Mikael Rinne

7

FRACOD3D: A Three-Dimensional Crack Growth Simulator Code Jingyu Shi and Baotang Shen

8

Coupled Fracture Modelling with RFPA . . . . . . . . . . . . . . . . . . . . . 173 Gen Li, Chun’an Tang, Zhengzhao Liang, and Lianchong Li

9

TOUGH-Based Hydraulic Fracturing Models . . . . . . . . . . . . . . . . . 203 Jonny Rutqvist

135

xv

xvi

Contents

10

Coupled Fracture Modelling with Distinct Element Methods . . . . . 227 Jeoung Seok Yoon and Jim Hazzard

Part III

Case Studies

11

Modelling Tunnel Failure and Fault Re-activation in CO2 Geo-sequestration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Baotang Shen, Nick Barton, and Jingyu Shi

12

Application to Underground LNG Storage . . . . . . . . . . . . . . . . . . . 285 Eui-Seob Park, Yong-Bok Jung, Taek Kon Kim, and Baotang Shen

13

Applications for Deep Geothermal Engineering . . . . . . . . . . . . . . . . 317 Linmao Xie, Bing Bai, Baotang Shen, Günter Zimmermann, and Ki-Bok Min

14

FRACOD Applications in Nuclear Waste Disposal . . . . . . . . . . . . . 347 Mikael Rinne

15

Mine Stability and Water Inrush in Coal Mine . . . . . . . . . . . . . . . . 361 Yunliang Tan, Shichuan Zhang, Weiyao Guo, Xuesheng Liu, and Baotang Shen

16

Modelling Hydraulic Fracturing in Coals . . . . . . . . . . . . . . . . . . . . 405 Weiguo Liang, Haojie Lian, and Jianfeng Yang

17

Applications of Rock Failure Process Analysis (RFPA) to Rock Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Chun’an Tang and Shibin Tang

18

Hydro-mechanical Coupled PFC2D Modelling of Fluid Injection Induced Seismicity and Fault Reactivation . . . . . . . . . . . . . . . . . . . 461 Jeoung Seok Yoon, Arno Zang, Hannes Hofmann, and Ove Stephansson

19

Lifetime Prediction of Rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 H. Konietzky, X. Li, and W. Chen

20

Numerical Simulation of Hydraulically Driven Fractures . . . . . . . . 531 Branko Damjanac, Christine Detournay, and Peter Cundall

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

Contributors

Bing Bai Institute of Rock and Soil Mechanics, Chinese Academy of Sciences (IRSM-CAS), Wuhan, China Nick Barton Nick Barton & Associates, Oslo, Norway W. Chen School of Civil Engineering, Central South University, Changsha, China Peter Cundall Itasca Consulting Group, Inc., Minneapolis, MN, USA Branko Damjanac Itasca Consulting Group, Inc., Minneapolis, MN, USA Christine Detournay Itasca Consulting Group, Inc., Minneapolis, MN, USA Melvin Diaz Korea University of Science and Technology, Daejeon, Republic of Korea Maurice Dusseault University of Waterloo, Waterloo, Canada Weiyao Guo Shandong University of Science and Technology, Qingdao, China Jim Hazzard Itasca Consulting Group, Minneapolis, MN, USA Hannes Hofmann Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences, Potsdam, Germany Sunggyu Jung Korea University of Science and Technology, Daejeon, Republic of Korea Yong-Bok Jung KIGAM, Daejeon, Republic of Korea Kwang Yeom Kim Korea Institute of Civil Engineering and Building Technology, Goyang, Republic of Korea Korea University of Science and Technology, Daejeon, Republic of Korea Taek Kon Kim SK E&C, Seoul, Republic of Korea

xvii

xviii

Contributors

Heinz Konietzky Geotechnical Institute, TU Bergakademie Freiberg, Freiberg, Germany Weiguo Liang Key Laboratory of In-situ Property-improving Mining of Ministry of Education, Taiyuan University of Technology, Taiyuan, Shanxi, China College of Mining Engineering, Taiyuan University of Technology, Taiyuan, Shanxi, China Zhengzhao Liang State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China Haojie Lian Key Laboratory of In-situ Property-improving Mining of Ministry of Education, Taiyuan University of Technology, Taiyuan, Shanxi, China College of Mining Engineering, Taiyuan University of Technology, Taiyuan, Shanxi, China Gen Li State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China Lianchong Li School of Resources and Civil Engineering, Northeastern University, Shenyang, China X. Li School of Resources and Safety Engineering, Central South University, Changsha, China Ki-Bok Min Department of Energy Resources Engineering, Seoul National University, Seoul, South Korea Xuesheng Liu Shandong University of Science and Technology, Qingdao, China Eui-Seob Park KIGAM, Daejeon, Republic of Korea Mikael Rinne School of Engineering, Department of Civil Engineering, Aalto University, Espoo, Finland Jonny Rutqvist Lawrence Berkeley National Laboratory, Energy Geosciences Division, Berkeley, CA, USA Baotang Shen CSIRO Mineral Resources, Brisbane, Queensland, Australia Shandong University of Science and Technology, Qingdao, China Jingyu Shi CSIRO Mineral Resources, Brisbane, Queensland, Australia Ove Stephansson GFZ German Research Centre for Geosciences, Potsdam, Brandenburg, Germany Royal Institute of Technology (KTH), Stockholm, Sweden Xizhen Sun School of Civil Engineering and Architecture, Linyi University, Linyi, China

Contributors

xix

Chun’an Tang State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China Shibin Tang State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, China Yunliang Tan Shandong University of Science and Technology, Qingdao, China Chin-Fu Tsang Lawrence Berkeley National Laboratory, Berkeley, CA, USA Uppsala University, Uppsala, Sweden Linmao Xie Department of Energy Resources Engineering, Seoul National University, Seoul, South Korea Jianfeng Yang Taiyuan University of Technology, Taiyuan, Shanxi, China Jeoung Seok Yoon DynaFrax UG, Potsdam, Germany Arno Zang Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences, Potsdam, Germany Baoliang Zhang Liaocheng University, Liaocheng, China Shichuan Zhang Shandong University of Science and Technology, Qingdao, China Li Zhuang Korea Institute of Civil Engineering and Building Technology, Goyang, Republic of Korea Günter Zimmermann Section 4.8 Geoenergy, German Research Center for Geosciences GFZ, Potsdam, Germany

Chapter 1

Introduction Baotang Shen, Ove Stephansson, and Mikael Rinne

Abstract The ability to predict and realistically reproduce rock mass behaviour using a numerical model is a pivotal step in solving many rock engineering problems. Several different types of numerical methods have been developed for various geomechanical problems. Although many existing numerical codes can model the behaviour of jointed or fractured rock mass, most do not consider the fracture initiation and propagation – a dominant mechanism, particularly in hard rocks. This book aims to introduce several unique numerical approaches to complex rock failure problems and demonstrate their capabilities through case studies and applications. The book contains 20 chapters which are broadly grouped into three parts. Part I describes the theoretical background of fracturing mechanics and latest understanding of rock fracturing mechanisms from laboratory tests and field tests. Part II describes the fundamentals of various numerical methods and codes for both basic fracture modelling and coupled processes. Part III describes a large number of case studies using various numerical codes for real rock engineering problems, including tunnel failure, fault re-activation, mine stability, water inrush, hydraulic fracturing, long term stability of rocks, and other key geomechanics issues related to EGS geothermal energy, CO2 Geosequestration, LNG underground storage, nuclear waste disposal and civil engineering.

Ove Stephansson died before publication of this work was completed.

B. Shen (*) CSIRO Mineral Resources, Brisbane, Queensland, Australia Shandong University of Science and Technology, Qingdao, China e-mail: [email protected] O. Stephansson GFZ German Research Centre for Geosciences, Potsdam, Brandenburg, Germany Royal Institute of Technology (KTH), Stockholm, Sweden M. Rinne School of Engineering, Department of Civil Engineering, Aalto University, Espoo, Finland e-mail: mikael.rinne@aalto.fi © Springer Nature Switzerland AG 2020 B. Shen et al. (eds.), Modelling Rock Fracturing Processes, https://doi.org/10.1007/978-3-030-35525-8_1

1

2

B. Shen et al.

It is our intension that this book will provide readers the latest developments in numerical methods and modelling tools for predicting the rock failure process involving explicit fracturing and thermal-hydraulic-mechanical coupling.

Rock mass is increasingly employed as the host medium for a vast array of human activities. Facilities like storage areas, wells, tunnels, underground power stations are located in a variety of rock types and under different rock mechanical conditions. Excavation stability is imperative for all such constructions, in both the short and long term. Understanding the long-term behaviour of a rock mass is crucial for safety and performance assessments of geological radioactive waste disposal. Hydro-thermalmechanical couplings of the ongoing processes around these repositories are particularly important. The understanding of fracturing of rock masses has also become a critical endeavour for energy extraction and storage. Small-scale breakouts around single wells in petroleum engineering can devastate the oil and gas extraction from source rock. The large-scale fracturing of rock formations for improved oil, gas and heat extraction is an essential field of development in the petroleum and geothermal industries. CO2 geosequestration is a complex new field of rock engineering where fracturing of the overburden rock during pressurization must be prevented while fracturing of the storage formation might be needed. All these intricate design tasks require powerful prediction and modelling tools. Failure of brittle rock is often associated with a rapid and violent event, as detected in short-term loading strength laboratory tests. From these test results, the mechanical properties of rock including fracture mechanics parameters are obtained. When rock is stressed close to its short-term strength, slow crack growth (also called subcritical crack propagation) occurs. With time, this slow fracturing process may generate critical stress concentrations that lead to a sudden unstable failure event. Slow subcritical crack growth (SCG) is thought to play an important role in longterm rock stability at all scales and for all kinds of rocks, ranging from laboratory samples to earthquake-generated faults. When sudden rock movement occurs in nature or around excavations the consequences can be catastrophic. The ability to predict and realistically reproduce rock mass behaviour using a numerical model is a pivotal step in solving many rock engineering problems. Numerical modelling can improve our understanding of the complicated failure processes in rock and the many factors affecting the behaviour of fractured rock. When our models manage to better capture the fundamental failure mechanisms observed in the laboratory, our ability to generate reliable large-scale models improves, as does our ability to predict the short- and long-term behaviour of rock masses in situ. Our ability to identify conditions where time is an important variable for the stability and long-term behaviour of rock excavations is likewise enhanced. Several different types of numerical methods have been developed for various geomechanical problems (Jing 2003). Since every method and code has its advantages and disadvantages, the choice of a suitable code should be carefully assessed for each rock-engineering problem. Code suitability depends on the character of the

1 Introduction

3

problem and the goal of the study. The mechanical behaviour of the rock mass is largely influenced by the presence of natural discontinuities like fractures, joints and faults. Hence, numerical methods that allow the introduction of displacement discontinuities into the continuous medium are often required in solving rock engineering problems. Numerical methods can be subdivided into “Continuum methods” and “Discontinuum methods”. Continuum methods (or continuum approaches) do not take into account the presence of distinct discontinuities. If natural discontinuities are numerous, then the substitution, at a certain scale, of a discontinuous medium with a continuous one is required. The mechanical characteristics of the continuous medium must be such that its behaviour is equivalent from a mechanical point of view to that of the discontinuous medium. The effects of fractures are smoothed out and the heavily jointed rock mass is considered as an equivalent continuous medium. The Discontinuum methods (or “Explicit joint approaches”) allow one to incorporate discrete discontinuities in the displacement field, that is, individual joints in the rock mass can be modelled explicitly. Discontinuum methods may describe the fracturing process using fracture mechanics principles. Although many existing numerical codes can model the behaviour of jointed or fractured rock mass, most do not consider the fracture initiation and propagation – a dominant mechanism, particularly in hard rocks. A very limited number of codes can model the fracture propagation but are usually not designed for application at engineering scales. This book aims to introduce several unique numerical approaches to complex rock failure problems and demonstrate their capabilities through case studies and applications. Much of the work described in this book is based on many years’ studies from an international collaboration project, named “Understanding and predicting coupled fracturing/fluid flow/thermal processes of jointed rocks in 2-D and 3-D” and the project was led and supported by CSIRO, Australia. This project was first initiated in 2007 with the original aim of developing the mechanical – thermal – hydraulic coupled functions in a fracture mechanics based code FRACOD. The objectives had been achieved during the first and second phases of the project during 2007–2011 and 2011–2016. As a result of this project, a book titled “MODELLING ROCK FRACTURING PROCESSES – A Fracture Mechanics Approach Using FRACOD” (Shen et al. 2014) was published which describes the details of the unique fracture mechanics based code FRACOD. Since 2016, the international collaboration project has moved into its 3rd phase and the scope of the project has been significantly widened. While the project still has its focus on FRACOD development and applications, other codes which have the capability in modelling rock fracturing processes (such as PFC, TOUGH, RFPA) are included in the study for the purpose of comparison and cross validation. More importantly, all these code developments and validations are ultimately for the real world rock engineering applications. Therefore, much of the focus in the project now has been applications and case studies using various numerical codes to solve engineering problems involving coupled rock fracturing processes. These latest researches form bulk of the contents in the book.

4

B. Shen et al.

The book contains 20 chapters which are broadly grouped into three parts. Part I (Chaps. 2, 3, 4 and 5) describes the theoretical background of fracturing mechanics and latest understanding of rock fracturing mechanisms from laboratory tests and field tests. 2D and 3D fracture propagation tests and hydraulic fracturing laboratory and field experiments are presented. Part II (Chaps. 6, 7, 8, 9 and 10) describes the fundamentals of various numerical methods and codes for both basic fracture modelling and coupled processes. It includes the latest developments in the two- and three-dimensional version of the fracture mechanics based code FRACOD, coupled fracture modelling with RFPA, TOUGH-based hydraulic fracturing models, and coupled fracture modelling with distinct element methods. Part III (Chaps. 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20) describes a large number of case studies using various numerical codes for real rock engineering problems. These case studies includes: • Modelling tunnel failure and fault re-activation of CO2 geosequestration using FRACOD • FRACOD modelling for LNG underground storage • Numerical applications to EGS geothermal energy • Mine stability and water inrush in coal mine • FRACOD applications to nuclear waste disposal • Modelling hydraulic fracturing in coal-bearing strata • Applications of Rock Failure Process Analysis (RFPA) to rock engineering • Coupled hydro-mechanical PFC modelling of fault reactivation and its application to EGS • Lifetime prediction of rocks • Simulation of hydraulic driven fractures The authors of each chapter are the leading experts in the relevant fields and have many years’ experience in numerical modelling and coupled processes. It is our intension that this book will provide readers the latest developments in numerical methods and modelling tools for predicting the rock failure process involving explicit fracturing and thermal-hydraulic-mechanical coupling.

References Jing L (2003) A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering. Int J Rock Mech Min Sci 40:283–353 Shen B, Stephansson O, Rinne M (2014) Modelling rock fracturing processes. A fracture mechanics approach using FRACOD. Springer, Dordrecht. ISBN 978-94-007-6903-8

Part I

Theoretical Background

Chapter 2

Introduction to the Theories of Rock Fracturing Mikael Rinne, Ove Stephansson, Baotang Shen, and Heinz Konietzky

Abstract This chapter provides the basic theories and principles behind rock fracturing and rock fracture mechanics. It starts with introducing the Griffith flaws and energy balance theory, which is the foundation of the modern fracture mechanics. Then the concept of stress intensity factor for linear elastic fracture mechanics is introduced, followed by a description of the criteria for fracture propagation. Also described is the subcritical crack growth which dominates the time dependent long term stability of fractured rock. The experience shows that rocks under static load level may be stable, while at the same cyclic load level failure is observed. A short introduction to theories and factors related to cyclic fatigue behavior of rocks is presented in the end of the chapter. Keywords Griffith’s flaws · Charles’ law · Stress intensity factor · Cyclic loading · Time dependency · Crack velocity · Energy balance theory · Fatigue behaviour

Ove Stephansson died before publication of this work was completed.

M. Rinne (*) School of Engineering, Department of Civil Engineering, Aalto University, Espoo, Finland e-mail: mikael.rinne@aalto.fi O. Stephansson GFZ German Research Centre for Geosciences, Potsdam, Brandenburg, Germany Royal Institute of Technology (KTH), Stockholm, Sweden B. Shen CSIRO Mineral Resources, Brisbane, Queensland, Australia Shandong University of Science and Technology, Qingdao, China e-mail: [email protected] H. Konietzky Geotechnical Institute, TU Bergakademie Freiberg, Freiberg, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Shen et al. (eds.), Modelling Rock Fracturing Processes, https://doi.org/10.1007/978-3-030-35525-8_2

7

8

2.1

M. Rinne et al.

Introduction

In rock mechanics, failure mechanisms generally refer to the overall processes of failure in a continuous sense, i.e. when a rock mass suffers damage permanently affecting its load-bearing ability. Failure criteria such as the well-known MohrCoulomb or Hoek-Brown criteria predict the failure conditions by applying the ultimate strength value of the rock matrix but do not account for any localised stresses caused by the discontinuities always inherent in the rock mass. Bieniawski (1967) distinguishes between a phenomenological failure criterion and a genetic failure mechanism. A phenomenological failure criterion simply provides a formula enabling the prediction of the strength values for all states of a multiaxial stress from a critical quantity that may be determined in one type of test, e.g. the uniaxial tensile or compression test. A genetic failure criterion describes the physical processes occurring in the material over the course of loading, eventually leading to failure. The failure of most brittle rock materials is essentially a process of crack initiation and propagation. Therefore the phenomenological failure criteria, although widely used as a good approximation of overall rock behaviour, may not be adequate to describe the failure in detail. The Griffith theory (1920), which led to the evolution of modern fracture mechanics, assumes that the propagation of the inherent flaws in the fabric is the source of failure of loaded brittle material. This approach provides a description of the transformation of an unbroken structural component into a fractured one by crack growth. Fracture mechanics assesses the strength of a stressed structure through the relationship between the loading conditions, the geometry of the crack and the resistance to crack propagation in terms of critical stress energy release rate (GC). Irwin proposed modification of Griffith’s theory (1957). He described the stress distribution around the crack tip and introduced the concept of the stress intensity factor (K). Irwin also showed the equivalence of strain energy release rate (G) and stress intensity factor (K). Irwin’s concept assumes that a fracture tip that has a stress intensity (K) equal to the material’s fracture toughness (KC – critical stress intensity factor) will accelerate to speeds approaching a terminal velocity that is governed by the speed of the elastic waves in the brittle medium. If the fracture propagation criterion is not met, i.e. if K < KC, then the fracture remains stable or shows only slow subcritical crack growth. In terms of the Griffith approach, a crack is stable when G < GC, where GC is the critical strain energy release rate. For a wide range of materials, it was determined experimentally that significant crack growth rates can occur at values of K or G often far below the critical values of these parameters. This phenomenon of subcritical crack growth (SCG) is one of the key mechanisms in the time dependent failure behaviour of rocks (Atkinson 1984). In this chapter crack initiation is the failure process by which one or more cracks are formed through micro-mechanical processes in previously macroscopically fracture-free material. Sometimes the crack initiation is referred to as the threshold stress value of the registered acoustic emission events σ ci (AE) for a structure under increasing load. The term crack initiation sometimes refers to the stress level where

2 Introduction to the Theories of Rock Fracturing

9

the volume of a test specimen in compression starts to increase after initial compaction. The term fracture propagation or crack propagation is defined as the failure process by which a pre-existing single crack or cracks extend or grow after fracture initiation. A distinction can be made between two types of crack propagation, namely ‘stable’ and ‘unstable’. Stable crack propagation is the process of fracture propagation in which the crack extension is a function of loading and can be controlled accordingly. Unstable crack propagation is the process where the crack extension is uncontrollable once started, which may be governed by loading and other factors, e.g. temperature and chemistry. A rupture is the failure process by which a structure (e.g. rock specimen) disintegrates into two or more pieces. A brittle fracture is defined as a fracture process exhibiting little or no permanent (plastic) rock deformation. In rock material under tension loading, fracture initiation, unstable fracture propagation and crack coalescence occur almost simultaneously. Crack coalescence describes when individual cracks merge and form a continuous fracture. Under compression, the friction between the two fracture surfaces must be considered. As a consequence the failure process in rock under compression is more complex compared to tensile failure. The rupture of the rock material results primarily from stable and unstable fracture propagation and crack coalescence rather than directly from fracture initiation. Rock materials are inhomogeneous and discontinuous at all scales. At the micro scale, defects (include micro-cracks, grain boundaries, pores, and bedding planes) causing stress concentrations. At the macro-scale, geological fractures are referred to as joints (open or closed) and faults (having signs of shearing) based on their genesis. The term ‘crack’ is reserved for short discontinuities of the grain size. A ‘fracture’ has a length more than several times the grain size and refers to a crack that has extended into the matrix. Although a fracture usually shows a more irregular trace compared with a short crack, from a modelling standpoint, cracks and fractures can be considered equivalent, but are of different scale.

2.2

Griffith Flaws and Energy Balance Theory

Given that the tensile strength of a material is generally much lower than theoretically predicted, Griffith (1920) postulated that typical brittle materials inevitably contain numerous randomly distributed sub-microscopic flaws, micro-cracks or other discontinuities. These discontinuities serve as stress concentrators and they are often referred to as Griffith flaws or cracks. Griffith established a relationship between critical stress and crack size, now known as the Griffith energy balance approach. It is the starting point for the development of modern fracture mechanics. Griffith explained how failure is caused by the extension of flaws or cracks in solid material. The creation of a new crack surface absorbs energy that is supplied from the work done by the external force. Release of the stored strain energy in the solid promotes the crack propagation and failure.

10

M. Rinne et al.

Fig. 2.1 A crack with the length 2a and width 2b subjected to (a) uniform tension; (b) biaxial compression. (After Whittaker et al. 1992)

The requirement for failure in solid material is both sufficient stress and sufficient energy. The stress requirement states that the local stress must be high enough to overcome the molecular cohesive strength. This can be achieved by stress concentrations due to the presence of discontinuities such as pre-existing micro-cracks. The energy requirement states that sufficient potential energy must be released to overcome the resistance to crack propagation. This can also be achieved through increasing the work done by external forces. The more energy a solid absorbs, the greater is its resistance to crack propagation. In pure tension, the total energy U of an infinite cracked plate (Fig. 2.1a) can be expressed in the following energy components (Whittaker et al. 1992): U ¼ Ut þ Uc  W þ Us

ð2:1Þ

U ¼ Total energy of the infinite cracked plate Ut ¼ Total initial elastic strain energy of the stressed but un-cracked plate Uc ¼ Total elastic strain energy release caused by the introduction of a crack of length 2a and the relaxation of material above and below the crack W ¼ Work done by the external forces Us ¼ Change in the elastic surface energy due to the formation of new crack surfaces (irreversible). The energy components can be obtained separately from the Theory of elasticity: U¼

σ 2 A πσ 2 a2 σεA þ 4aγ s   2 2E 0 E0

ð2:2Þ

σ ¼ Tensile stress a ¼ Half-crack length E0 ¼ Effective Young’s modulus: E0 ¼ E for plane stress and E’ ¼ E/(1-ν2) for plane strain ν ¼ Poisson’s ratio A ¼ The infinite area of the thin plate of unit thickness (B ¼ 1) ε ¼ Average axial strain in the plate γS ¼ Specific surface energy, i.e. energy required to create a unit area of new crack surface as the crack increases in length.

2 Introduction to the Theories of Rock Fracturing

11

Fig. 2.2 Load-displacement illustration for crack propagation: (a) constant displacement; (b) constant load. (After Whittaker et al. 1992)

Two extreme loading conditions can be considered (Fig. 2.2): (a) Constant displacement (fixed-gripped loading) where the applied loading system suffers zero displacement as the crack extends; (b) Constant load (dead-weight loading) where the applied force remains constant as the crack extends. In the first case there is no work done by the external force P. Therefore, Uc should be negative in Eq. (2.2). In the second case, the work done by the applied load P increases the elastic strain energy release and accordingly Uc should be positive. Boundary conditions in real loading situations are generally somewhere between case (a) and (b). Note that W must be subtracted from the reversible energy terms, since it does not form part of the plate’s potential energy (Up ¼ Ut + Uc  W). The mechanical energy released during incremental crack propagation is also independent of the loading configuration. Griffith’s idea implies that the critical equilibrium for fracture initiation occurs when: ∂U ¼ 0, ∂a

ð2:3Þ

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ π  a ¼ 2E 0  γ s

ð2:4Þ

leading to:

Equation (2.4) indicates that fracture initiation in ideally brittle solids is governed by the product of an applied far-field stress and the square root of the crack length and by the material properties characterized by effective Young’s modulus E’ and the specific surface energy γs. Rearrangement of Eq. (2.4) gives: πσ 2 a ¼ 2γ s E0

ð2:5Þ

12

M. Rinne et al.

The left-hand side of the equation represents the elastic energy per unit crack surface. This energy is available for crack propagation and it defines an important parameter called the strain energy release rate, denoted by G in honour of Griffith. Thus, an expression for the strain energy release rate G can be given as: G¼

πσ 2 a E0

ð2:6Þ

Griffith’s theory for the critical condition for fracture initiation becomes: G ¼ GC

ð2:7Þ

where GC is the critical strain energy released per unit length of crack extension. More generally, G is defined as the derivative of the elastic strain energy release with respect to crack area rather than crack length. The above formulation describes the simple case of a flat and open elliptical crack subjected perpendicularly to a uniaxial tensile load, assuming that the crack propagates along its own plane. As indicated, the strain energy release rate is the governing parameter for crack initiation, which is accordingly referred to as the energy balance approach. With his flaw hypothesis and energy balance concept Griffith laid a solid foundation for a general theory of fracture. He further improved his concept known as the ‘fracture stress approach’ to consider more complicated stress fields involving compression (Fig. 2.1b). The energy change of crack formation can be considered as entire crack formation from the initially intact rock body, as presented above, or as an incremental extension of an existing crack. In Fig. 2.1a the boundary conditions were obtained as uniform far-field tensile stresses. In practical applications it is often useful to describe crack extension in terms of an external force (point load P), a crosssection of the new crack surface (dc) and elastic compliance (λ), defined as the load-point displacement (u0) per unit load (e.g. Lawn 1993). Parameter G can be evaluated with respect to new crack area rather than the crack length at equilibrium for crack initiation (Eq. 2.3).

2.3

Loading Modes and Associated Displacements

The flat crack tip in an ideally linear elastic brittle material can be subjected to a normal stress σ, an in-plane shear stress τi, an out-of-plane (or anti-plane) shear stress τo, or any combination of these. Figure 2.3 illustrates the crack tip coordinates and stress state in terms of both cartesian and polar coordinates. Different loading configurations at the crack tips lead to different modes of crack tip surface displacements. Three basic loading configurations form the fracture modes of crack tip deformation: Mode I, II and III, as illustrated in Fig. 2.4.

2 Introduction to the Theories of Rock Fracturing

13

Fig. 2.3 Crack tip coordinates and stress state in Cartesian and polar coordinate systems. (After Whittaker et al. 1992)

Fig. 2.4 The three basic modes of loading for a crack and the corresponding crack surface displacements. (After Whittaker et al. 1992)

Mode I is the opening (or tensional) mode. The crack tip is subjected to a stress σ normal to the crack plane and crack faces separate at the crack front so that the displacements of the crack surfaces are perpendicular to the crack plane. Mode II is the edge-sliding mode (or in-plane shearing) where the crack tip is subjected to an in-plane shear stress τi and crack faces slide relative to each other so that the displacements of the crack surfaces are on the crack plane and perpendicular to the crack front. Mode III is the tearing (or out-of-plane shearing) mode. The crack tip is subjected to an anti-plane shear stress τo. The crack faces move relative to each other so that the displacement of the crack surfaces are in the crack plane but parallel to the crack front. Mixed-mode loading is a combination of any of the three loading modes. For example, a combination of Mode I and Mode II loading forms a Mixed-mode I–II loading.

14

2.4

M. Rinne et al.

Stress Intensity (K) Approach

Griffith’s energy balance approach formed a solid basis for Irwin’s (1957) widely applied ‘stress intensity factor approach’. This is based on the crack tip characteristic parameter, called the stress intensity factor K, which uniquely governs the crack tip stress and displacement fields. For a given cracked body under a certain type and magnitude of loading, K is known and the stresses and displacements can be determined accordingly. The three stress intensity factors (KI, KII, KIII) each correspond to a loading mode (Fig. 2.4), and each is associated with a unique stress distribution near the fracture tip. A detailed stress state for an infinite plate containing a central crack of length 2a under uniaxial tension, σ, as shown in Fig. 2.5a (Mode I) is given: 3 θ 3θ 1  sin sin 6 2 2 7 σx 7 6 KI θ6 7 6 7 θ 3θ 7 4 σ y 5 ¼ pffiffiffiffiffiffiffiffiffiffiffi cos 6 1 þ sin sin 2 6 2 2 7 2π  r 5 4 σ xy θ 3θ sin cos 2 2

ð2:8Þ

pffiffiffiffiffiffiffiffiffi KI ¼ σ π  a

ð2:9Þ

2

2

3

Where

and r is the distance from crack tip, σ z ¼ ν(σ x + σ y) for plane strain and σ z ¼ σ xz ¼ σ yz ¼ 0 for plane stress. The crack tip stress components are often expressed in terms of polar coordinates in 2D: 3 2θ 1 þ sin 6 2 7 σr 7 6 KI θ6 7 6 7 θ 7 4 σ θ 5 ¼ pffiffiffiffiffiffiffiffiffiffiffi cos 6 cos 2 2 7 6 2 2π  r 5 4 σ rθ θ θ sin cos 2 2 2

3

2

ð2:10Þ

where σ z ¼ ν(σ r + σ θ) for plane strain and σ z ¼ σ rz ¼ σ θz ¼ 0 for plane stress. The stress intensity factors K for Mode II and Mode III can be similarly defined as follows: pffiffiffiffiffiffiffiffiffi K II ¼ τi π  a and

ð2:11Þ

2 Introduction to the Theories of Rock Fracturing

15

Fig. 2.5 An infinite plate containing a crack under biaxial tension, in-plane shear and anti-plane shear. (After Whittaker et al. 1992)

pffiffiffiffiffiffiffiffiffi K III ¼ τ0 π  a

ð2:12Þ

where τi is the far-field in-plane shear stress and τo is the far-field anti-plane shear stress (Fig. 2.5). The stress intensity factors (KI, II, III) are dependent on the magnitude of the far-field stress, the crack size and the loading conditions. In this sense, stress intensity factors can be physically regarded as fracture parameters reflecting the distribution of the stress in a cracked brittle body. Consequently, for any specific mode, with knowledge of the stress intensity factor, the crack tip stresses and displacements can be determined. The derivation of crack tip displacements follows the crack tip stresses using Hooke’s law and theory of elasticity. According to the superposition principle, crack tip stress and displacement components for a Mixedmode I–II loading can be obtained by superimposing those resulting from pure Mode I and pure Mode II loadings, and likewise for other loading combinations. Closed-form solutions of the stress and displacement functions can be found for simple loading configurations in fracture mechanics handbooks (Whittaker et al. 1992; Lawn 1993). In practice, the geometries of cracked bodies and loading conditions are usually complicated, so closed-form solutions are not generally obtainable. For a cracked body of finite dimensions, numerical methods are usually needed to calculate the stress and displacement distributions. The problem is further compounded when fracture criteria are used to study crack initiation and propagation under compression because the crack faces tend to close and frictional forces must be considered. According to Eqs. (2.8) and (2.10), the stress approaches infinity (singular) at the tip of a crack (when r ! 0). However, this is practically impossible since no material can bear infinite stress. When the stresses near the crack tip exceed the yield strength σ ys the material yields until the stresses drop below σ ys. Accordingly, a small region around the crack tip is formed in which the material behaves plastically rather than elastically as is usually assumed for the treatment of fracture mechanics problems. This small region has many names depending on the material. For brittle rock, this region is called the ‘crack tip micro-cracking zone’, the ‘crack tip inelastic zone’ or

16

M. Rinne et al.

Fig. 2.6 Idealized pictures of mode I, II and III fracture propagation in LEFM (top) and NEFM (bottom). An idealized, sharp cut in elastic, homogeneous, isotropic material represents (a) mode I tensile crack, (b) mode II in-plane shear crack and (c) mode III anti-plane shear crack growth in LEFM. The idealized shape of the process zone of (d) tensile fracture, (e) sliding mode shear fracture and (f) tearing mode shear fracture is contoured by approximate solutions in NEFM. (Zang and Stephansson 2010)

the ‘Fracture Process Zone (FPZ)’ (Fig. 2.6). The presence of this inelastic zone ahead of the crack tip affects the fracture behaviour of the material. The application of an elastic analysis to a real cracked body depends on the extent of the FPZ. If the FPZ is sufficiently small compared with the geometry of the crack and any other characteristic dimensions of the specimen, then it can be assumed that the linear elastic behaviour prior to failure prevails. If the inelastic zone satisfies the requirement, it is referred to as small-scale yielding (SSY) and the elastic analysis of such a cracked body is termed Linear Elastic Fracture Mechanics (LEFM). In general, conditions for SSY can be met for a number of materials like brittle rocks, concrete, glasses and ceramics. In contrast, if the non-linear elastic deformation is a dominant preceding failure and the non-linear elastic zone is substantial, i.e. large-scale yielding (LSY), then such an analysis is termed Non-Linear Elastic Fracture Mechanics (NEFM). In such analyses, K can no longer characterise the crack tip stress, strain and displacements, since prior to failure the crack tip has become very blunted due to the formation of a yielded region in metals or micro-cracking in rocks. The R-curve concept, developed from the Griffith energy balance theory, can be used to address fracture problems

2 Introduction to the Theories of Rock Fracturing

17

involving the crack tip non-linear region. J-integral and crack opening displacement approaches can treat fracture problems involving relative large crack tip non-linear zones.

2.5

Relationship Between G and K

Irwin (1957) showed the relation between the global energy parameter, the strain energy release rate G, and the local crack tip parameter, the stress intensity factor K: GI ¼

K 2I E0

ð2:13Þ

GII ¼

K 2II E0

ð2:14Þ

GIII ¼

K 2III 2μ

ð2:15Þ

where GI, GII and GIII are the strain energy release rates for Mode I, Mode II and Mode III, respectively, E’ is the effective Young’s modulus (see Eq. 2.2) and μ is the shear modulus. μ¼

E 2ð 1 þ ν Þ

ð2:16Þ

The relationships between G and K for different modes of loading are obtained by assuming that the crack extends along its own plane. If a crack extends at an angle with respect to the crack plane, the relation between G and K is more complex. When a crack is exposed to a Mixed-mode I–II loading, the overall strain energy release rate, G, is a summation of the Mode I strain energy release rate, GI, and that of Mode II, GII, which indicates that the strain energy release rates for various loading modes are additive and that the superposition principle applies not only to the same mode but also to different modes. This is similar to the crack tip stress and displacement fields, but unlike the stress intensity factors that are additive only for the same mode (Whittaker et al. 1992). The equivalence of G and K is important and forms the basis for the development of other branches of fracture mechanics involving LEFM, NEFM, dynamic fracture mechanics, statistical fracture mechanics, composite fracture mechanics etc.

18

2.6

M. Rinne et al.

Stress Intensity Factor K and the Critical Value KC

If the fracture initiation is expressed for Mode I loading in terms of the stress intensity factor, then crack initiation occurs when the stress intensity factor reaches its critical value called the Mode I plane strain fracture toughness, KIC: K I ¼ K IC

ð2:17Þ

Similarly, in Mode II, when the crack tip stress intensity factor, KII, reaches the Mode II plane strain fracture toughness, KIIC, cracking will initiate. Fracture toughness is basically a property of the material reflecting its resistance to physical macroscopic separation through crack propagation. Conceptually, KC is a constant, and since KC can be obtained by laboratory testing with specified methods and using specimens with known corresponding stress intensity factors, this approach has gained popularity. Analyses related to Mixed-mode loading conditions are common. For example, for an angled crack subjected to a uniform far-field compressive stress, both KI and KII at the crack tip must be considered (Fig. 2.1b). According to the superposition principle, the crack tip stress and displacement components can be obtained by superimposing those resulting from pure Mode I and pure Mode II loadings. Crack propagation will occur when a certain combination of KI and KII, f(KI, KII), reaches a critical value, f(KI, KII)C. The quantity f(KI, KII)C is known as the Mixed-mode I–II fracture toughness envelope or the KI–KII envelope. The question is, what is the exact KI–KII envelope as a criterion for Mixed-mode I–II cracking? The development of a fracture criterion to predict the initiation and propagation of individual cracks in rock subjected to arbitrary loading conditions is of importance for rock engineering. With the assumption that crack propagation is governed by a specific parameter, various fracture criteria have been established. The three fundamental fracture criteria appear to be the most frequently cited approaches in the literature: maximum tangential stress, maximum energy release rate and minimum strain energy density. These criteria can all predict the propagation and direction of crack initial extension under Mixed-mode I–II loading. Descriptions of a number of Mixed-mode fracture criteria and comparisons of predicted results are presented in Whittaker et al. (1992), Shen (1993) and Rao (1999). Shen and Stephansson (1993) suggest a criterion for fracture propagation under Mixed-mode I–II loading based on the maximum energy release rate.

2.7

Crack Velocity

The time dependency of crack growth is due to rate-controlled processes acting at the tips of cracks where stress concentration exists. Propagation velocities can vary over many orders of magnitude as a function of the stress intensity. Experimental studies on rocks have been made at crack velocities down to 109 m/s and K values less than 0.5 KC (Atkinson and Meredith 1987).

2 Introduction to the Theories of Rock Fracturing

19

Slow crack velocity is due to subcritical crack growth, SCG. Subcritical crack growth can be caused by several competing mechanisms, such as stress corrosion, diffusion, dissolution, ion exchange and micro-plasticity. One particular mechanism will be dominant under specific ranges of environmental and material conditions. In its most elementary form, the theory of stress corrosion postulates that for crystalline silicates and for silicate glasses the strained Si-O bonds at crack tips can react more readily with environmental agents than can unstrained bonds, because of a straininduced reduction in the overlap of atomic orbital. SCG velocity increases as (G or K ) is increased. The exact form of the relationship is v ¼ f ðG or K Þ

ð2:18Þ

and it depends on the crack growth mechanisms, until the critical value (G or K )c is achieved. At this critical level, the crack propagates rapidly accelerating to speeds approaching a terminal velocity that is governed by the speed of the elastic waves. Figure 2.7 shows the three regions of behaviour from SCG studies on glass. The SCG mechanisms and the interaction between the environment and the microstructure of the solid will control the details of the stress intensity factor/crack velocity curve. The schematic figure is presented merely as a starting point and its shape varies considerably when the effective factors such as temperature, pore pressure, pH, etc., are changed (Atkinson 1984). The behaviour in Region 1 is controlled by the rate of stress corrosion reactions at the crack tips. Region 2 is controlled by the rate of transport of reactive species to the crack tips. In Region 3, crack growth is mainly controlled by mechanical rupture and is relatively insensitive to chemical environment. Most experimental data on subcritical tensile crack propagation in geological materials appear to be in Region 1 or Region 3 of the schematic stress intensity factor/crack velocity curve. Region 2 is observed infrequently in rocks, although apparently it was found in tests on black gabbro in water (Atkinson 1984). It is assumed that a threshold exists below which no significant crack propagation can occur through stress corrosion (K0, stress corrosion limit). The value of this parameter is a function of the material’s fracture properties and environment. It is likely that K0 is a small fraction of KC, about 10–20%. However, experiments have not yet confirmed the existence of a stress corrosion limit in ceramics and rocks (Atkinson 1984).

2.8

Charles’ Law and Crack Tip Velocity

Charles explored the delayed failure of glass in relation to its sensitivity to atmospheric corrosion (1958). His study investigated the rate of corrosion layer formation of lime glass rods treated in saturated water vapour. An analysis of the failure process was presented based on the concept that inherent surface flaws grow by corrosive mechanisms to critical dimensions through a reaction between atmospheric

20

M. Rinne et al.

Fig. 2.7 Schematic stress intensity factor/crack velocity diagram for subcritical tensile crack on glass. (After Atkinson 1984)

water vapour and the components of glass. The rate of this reaction is determined by local stress conditions and by the temperature, pressure and composition of the surrounding atmosphere. Experimental work shows a close relationship between the temperature dependence of the failure process and that of the self-diffusion of sodium ions in bulk glass. It is concluded that the alkali content is responsible for the very low long-term strengths of most inorganic glasses. Charles assumed a stress power relationship to describe the crack velocity controlled by the rate of stress corrosion reactions at the crack tip: vðT Þ ¼ k0 ðσ m Þn þ k

ð2:19Þ

where v ¼ Penetration velocity of crack tip T ¼ Temperature σm ¼ Tensile stress at crack tip. k ¼ Corrosion rate of the material under zero stress, and k0 and n are constants. For stress-activated corrosion, Charles further assumed that the temperature dependence of the flaw growth process takes the form of an Arrhenius-type relationship. Bearing in mind that the stress at the crack tip is related to crack size and geometry, he suggested the following crack growth relationship: 

x vC xcr

n=2

where C ¼ Constant x, xcr ¼ Crack size and critical crack size.

eA=RT

ð2:20Þ

2 Introduction to the Theories of Rock Fracturing

21

A ¼ Activation energy term, and R ¼ Gas constant. A variety of mathematical functions can be fitted to the laboratory data when describing the stress intensity as a function of subcritical crack velocity. Subcritical crack growth data can be expressed with the power law, exponential and hyperbolic functions. Examples of fracture mechanics and subcritical crack growth approach to model time dependent failure of rocks can be found in Shen and Rinne (2007) and in Rinne (2008).

2.9

Cyclic Fatigue Behaviour of Rocks

Except static loading also cyclic loading (load reversal) is an important issue in geotechnical engineering, e.g. for bridge crossing, industrial or traffic vibrations, drilling or cutting induced vibrations, seismic loading, wind and wave loading, cyclic fatigue hydraulic fracturing, freeze-thaw cycles, cyclic filling and depletion of reservoirs in petroleum engineering, explosions or blasting. The experience shows that objects exposed to the same load level have longer life time under static loading compared to cyclic loading. Also, it becomes apparent, that rocks under static load level may be stable, while at the same cyclic load level failure is observed. Different factors have influence on the cyclic fatigue behavior of rocks (Cerfontaine and Collin 2018): frequency, maximum stress, stress amplitude, confinement, degree of saturation, anisotropy, waveform and sample size. Typical lab tests to investigate the cyclic fatigue behavior of rocks are: cyclic uniaxial and triaxial compressive tests, cyclic indirect (Brazilian) tensile tests, and cyclic 3- and 4-point bending tests. The degree of fatigue is typically characterized by the following damage variables (Song et al. 2018a, b; Cerfontaine and Collin 2018): residual deformation (axial or volumetric), wave velocity, deformation modulus, AE count or energy, dissipated energy/energy ratio and permeability. The most classical and still often used representation of the cyclic fatigue behavior is the so-called S-N (Wöhler) curve, which results in a straight line if a semi-logarithmic plot is used. The S-N curve relates the number of cycles N up to failure to the normalized ratio of maximum cyclic stress σmax to monotonic strength σmon. A and B are material parameters in Eq. (2.21): σ max ¼ A  log 10 N  B σ mon

ð2:21Þ

Besides phenomenological approaches, also fracture mechanical based approaches can be used. For a sinusoidal excitation the parameters shown in

22

M. Rinne et al.

Fig. 2.8 Basic fracture mechanical parameters for cyclic loading

Fig. 2.9 Phases of crack propagation under cyclic loading

Fig. 2.8 are given, where N is defined as the number of cycles and K as the stress intensity factor. According to Fig. 2.8 the following definitions are valid: ΔK ¼ K max  K min K max þ K min Km ¼ 2 K min R ¼ K max

ð2:22Þ

The increment in fracture length per loading cycle (da/dN ) can be represented in a logarithmic plot (Fig. 2.9), where ΔKth represents the stress intensity factor magnitude, below which no crack propagation occurs and ΔKc represents the critical stress intensity factor magnitude, where critical crack propagation starts. Three phases can be distinguished: • Phase I: decelerating phase • Phase II: stationary phase • Phase III: accelerating phase

2 Introduction to the Theories of Rock Fracturing

23

Crack propagation within phase I can be described by the “Donahne” law: da ¼ K ðΔK  ΔK th Þm dN

ð2:23Þ

where: ΔK th ¼ ð1  RÞγ ΔK th,

0

ð2:24Þ

ΔKth 0 is the threshold for R ¼ 0 and γ is a material parameter. Crack propagation within phase II is given by the “Paris-Erdogan“relation: da ¼ C ðΔK Þm dN

ð2:25Þ

m and C are material constants, where m is often set to 4. Crack propagation within phase III can be described by the “Forman” law: C  ðΔK Þn da ¼ dN ð1  RÞ K c  ΔK

ð2:26Þ

C and n are material constants. An expression, which covers all three phases is the so-called “Erdogan and Ratwani” law: C ð1 þ βÞm  ðΔK  ΔK th Þn da ¼ dN K c  ð1 þ βÞ ΔK

ð2:27Þ

where β ¼

K max þ K min K max  K min

ð2:28Þ

and c, m, n are material constants. Different techniques to simulate static fatigue, is discussed in more detail in Chap. 19. For numerical simulation of cyclic fatigue see for instance Song et al. (2018a, b).

24

M. Rinne et al.

References Atkinson BK (1984) Subcritical crack growth in geological materials. J Geophys Res 89:4077–4114 Atkinson BK, Meredith PG (1987) The theory of subcritical crack growth with applications to minerals and rock. In: Atkinson BK (ed) Fracture mechanics of rock. Academic, London, pp 111–162 Bieniawski ZT (1967) Mechanics of brittle fracture of rock. Part I, II and III. Int J Rock Mech Min Sci 4(3):95–430 Cerfontaine B, Collin F (2018) Cyclic and fatigue behaviour of rock material: review, interpretation and research perspective. RMRE 51(2):391–414 Charles RJ (1958) Static fatigue of glass. J Appl Phys 1958(29):1549–1560 Griffith AA (1920) The phenomena of rupture and flow in solids. Philos Trans Royal Soc Lond Ser A 221:163–198 Irwin GR (1957) Analysis of stresses and strains near the end of a crack. J Appl Mech 24:361–364 Lawn B (1993) Fracture of brittle solids, 2nd edn. Cambridge University Press, Cambridge Rao Q (1999) Pure shear fracture of brittle rock – a theoretical and laboratory study. PhD thesis 1999:08, Lulea University of Technology Rinne M (2008) Fracture mechanics and subcritical crack growth approach to model timedependent failure in brittle rock. Doctoral dissertation, Helsinki University of Technology. Faculty of Engineering and Architecture. Department of Civil and Environmental Engineering. Available on http://lib.tkk.fi/Diss/2008/isbn9789512294350/ Shen B, Rinne M (2007) A fracture mechanics code for modelling sub-critical crack growth and time dependency. In: Eberhardt E, Stead D, Morrison T (eds) Proceedings of the 1st Canada-US rock mechanics symposium, Vancouver, pp 591–598 Shen B (1993) Mechanics of fractures and intervening bridges in hard rocks. Doctorate thesis, Royal Institute of Technology, ISBN 91-7170-140-0 Shen B, Stephansson O (1993) Modification of the G-criterion of crack propagation in compression. Int J of Eng Fract Mech 47(2):177–189 Song Z, Konietzky H, Frühwirt T (2018a) Hysteresis energy-based failure indicators for concrete and brittle rocks under the condition of fatigue loading. Int J Fatigue 114:298–310 Song Z, Konietzky H, Herbst M (2018b) Bonded-particle model-based simulation of artificial rock subjected to cyclic loading. Acta Geotechnica. Springer-Verlag GmbH, Germany Whittaker BN, Singh RN, Sun G (1992) Rock fracture mechanics. Principles, design and applications, Developments in geotechnical engineering, 71. Elsevier, Amsterdam Zang A, Stephansson O (2010) Stress field of earth’s crust. Springer. ISBN 978-1-4020-8443-0, e-ISBN 978-1-4020-8444-7

Chapter 3

Laboratory Studies of 2D and 3D Rock Fracture Propagation Baotang Shen, Xizhen Sun, and Baoliang Zhang

Abstract To study mechanisms of crack propagation and coalescence in rock likematerials, several experimental studies were performed using gypsum specimens with pre-existing 2D and 3D cracks under uniaxial and true tri-axial compression loading. Video camera, Acoustic Emission monitoring and Computed Tomography scan had been used to assist the observation and interpretation of crack propagation processes. Results show that cracks in rock-like material can propagate and coalesce by shear failure or tensile failure. The propagation path mainly depends on the loading condition and the geometry and configurations of the pre-existing cracks. Further laboratory study was carried out to investigate the propagation characteristics of edge cracks, which simulates the situation of underground longwall mining. The mechanism of rock fracturing around the edge crack was investigated, and research findings have a major implication to water inrush prevention in underground coal mines. Keywords 2D and 3D cracks · Crack propagation · Uniaxial loading · Tri-axial loading · Mode I · Mode II

B. Shen (*) CSIRO Mineral Resources, Brisbane, Queensland, Australia Shandong University of Science and Technology, Qingdao, China e-mail: [email protected] X. Sun School of Civil Engineering and Architecture, Linyi University, Linyi, China e-mail: [email protected] B. Zhang Liaocheng University, Liaocheng, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Shen et al. (eds.), Modelling Rock Fracturing Processes, https://doi.org/10.1007/978-3-030-35525-8_3

25

26

3.1

B. Shen et al.

Introduction

The initiation, propagation and coalescence of rock cracks is commonly recognised to govern mechanical behaviour of brittle rocks. Numerous experimental and analytical investigations on microcrack initiation, growth and coalescence have been conducted since Griffith (1921) proposed the mechanism, see e.g. Kachanov (1982), Kemeny and Cook (1986), Wong (1982). Studies on cracked rock not only help to explain crack propagation mechanism, but also serve as models for the behavior of jointed (fractured) rock masses. Joint extension and coalescence can reduce the stiffness of jointed rock masses (Shen and Stephansson 1992), cause the shear failure of rock slopes (Einstein et al. 1983), and induce earthquakes by forming shear faults (Deng and Zhang 1984). Coalescence of cracks has been investigated by many researchers both experimentally and analytically. Lajtai (1969) performed direct shear tests on natural rock specimens with two parallel slots, Segall and Pollard (1980) investigated analytically the stress field in rock bridges between two stepped cracks, Horri and Nemat-Nasser (1986) studied the coalescence behaviour of multicracks in polymer specimens, and Reyes and Einstein (1991) conducted uniaxial tests on gypsum specimens with two inclined flaws. These studies have increased our knowledge about the behaviour of multi-fractured rock. However, most of the previous studies, especially the experimental studies, were carried out by using pre-existing slots and flaws whose surfaces were not in contact. Hence, we do not know under what conditions cracks or fractures with surface contact and friction propagate and coalescence. The answer to this question is of special significance for the understanding of the failure of rock masses, where joints or other discontinuities have surface contact and friction in most situations. Rock fractures are three-dimensional, hence the study of 3D cracks initiation and propagation is important for understanding the rock mass behaviour in rock engineering applications. Several previous experiments were conducted to study 3D crack mechanisms on specimens containing pre-existing 3D cracks under compression loads (Dyskin et al. 1999; Dyskin et al. 2003; Wong et al. 2004; Huang and Wong 2007; Guo et al. 2008; Lin et al. 2008; Li and Lu 2012; Li et al. 2012). The 3D cracks propagation experiments in glass materials containing single flaw and two flaws were studied by Dyskin et al. (1999, 2003); the mechanisms of elliptical cracks, pennyshaped cracks and 3D surface cracks were investigated systematically on PMMA (Wong et al. 2004; Huang and Wong 2007); the propagation of pre-existing 3D cracks in composite materials was studied under tri-axial loading by Lin et al. (2008); the propagation of 3D parallel crack groups was observed on unsaturated resin materials under uniaxial loading by Guo et al. (2008); real-time CT scan technique was used to investigate the propagation of 3D penny crack sets in ceramic materials under tri-axial loading condition by Li and Lu (2012); the mechanisms of 3D cracks were investigated under biaxial loading by Li et al. (2012). The previous experimental results showed that tensile cracks were the main propagation mode for 3D cracks

3 Laboratory Studies of 2D and 3D Rock Fracture Propagation

27

under compression and the tensile cracks were initiated from the tips of pre-existing cracks and propagated in the direction of the maximum principal stress. Shear cracks and secondary cracks were hardly observed in the entire experimental process. Although these previous studies have provided a good understanding of 3D crack propagation in densely compacted materials which are transparent for real-time observation, shear cracks and secondary cracks were difficult to be observed in experiments (Shen 1993). Mode II (shear) fractures are often initiated with shear dilation effect. In rocks or rock-like materials which contain pores, flaws and other defects, the volume expansion caused by shear failure could be absorbed by these existing voids in the material, preventing a major normal stress increase; however, in densely compacted materials such as glass and PMMA, shear dilation effect would lead to a significant localized normal stress increase, which can lead to high frictional resistance and prevent shear crack propagation. This chapter focuses on the experimental studies conducted on 2D frictional cracks and 3D cracks in rock-like materials. It includes: (1) uniaxial compression tests on 2D frictional cracks; (2) triaxial compression tests on 3D cracks; (3) uniaxial compression tests of edge cracks. Video camera, Acoustic Emission (AE) monitoring and CT scan had been used to assist the observation and interpretation of crack propagation processes.

3.2

Uniaxial Compression Tests on 2D Frictional Cracks

In this study, we conducted a series of uniaxial loading tests on gypsum specimens containing two cracks. Two types of cracks were used: (i) cracks without surface contact and friction, namely open cracks, and (ii) cracks with surface contact and friction, namely closed cracks. The results from testing specimens with open cracks and closed cracks were compared. Cracks with different configurations (inclination of cracks and inclination of the rock bridge between the cracks) were investigated. The failure process was monitored with a microscope and video camera and the failure mechanism is evaluated. Additional details of this study can be found in Shen et al. (1995).

3.2.1

Specimen Preparation and Testing

Specimens were made from gypsum, water and celite at ratios gypsum/water/ celite ¼ 165/70/2 in weight. This mixture had been previously reported to behave similarly to brittle rock (Nelson 1968). The size of the prismatic specimens was 152.4  76.2  30 mm (Fig. 3.1a). Two cracks with lengths of 12.7 mm each were created in the center of the specimens during casting (Fig. 3.1b). The length of the

28

B. Shen et al.

30 mm

Outer tip

Inner tips

152.4 mm

-- Bridge inclination

-- Crack inclination

76.2 mm

(a)

(b)

Fig. 3.1 Geometry of specimens and pre-existing cracks

bridge between the two inner tips was also 12.7 mm. Since the length of the cracks was much smaller than the thickness of the specimen, the cracks can be considered to be in the plane strain condition. Different cracks and bridges inclinations were used to investigate the influence of crack geometry on failure mode and failure load. Open cracks in the specimen were created by positioning two thin (0.04 mm) steel sheets in the mould before casting and pulling them out when the gypsum mixture became hard. Closed cracks were created using the method as follows: we pre-installed two tightly stretched polyethylene sheets (thickness 10μm) in the desired crack positions before casting. After casting and 30 minutes of curing when the gypsum mixture started to become hard, the polyethylene sheets were pulled out. The expansion of the gypsum mixture closed the thin slots and hereby created the closed cracks. Additional loading tests of small specimens showed that the closed cracks made in this way have approximately a friction angle of 35 and a cohesion of 3.5 MPa. The uniaxial tests were performed in an INSTRON (Model 1331) servocontrolled hydraulic loading machine. Displacement control was used to avoid a sudden failure of the specimens. The crack initiation and propagation were monitored by a microscope connected to a video recorder. The whole process of failure was recorded and studied by re-playing it in slow motion (the coalescence process was usually very fast and could not be seen by eyes). Simultaneously, the load and displacement were recorded and the complete loading and unloading curve was drawn. Figure 3.2 shows a schematic loading and monitoring system.

3 Laboratory Studies of 2D and 3D Rock Fracture Propagation Fig. 3.2 Loading and monitoring system of the crack coalescence test. (1) load cell; (2) operating and recording device; (3) specimen with pre-existing cracks; (4) microscope; (5) video recorder; (6) TV

29

1

2 4

6

3

5

Wing cracks Secondary cracks

Coalescence

Pre-existing cracks

(a)

(b)

(c)

Fig. 3.3 Typical sequence of crack initiation, propagation and coalescence. (a) Wing cracks initiated from the pre-existing crack tips at low uniaxial load; (b) wing cracks propagated with increasing load and secondary cracks appeared at the pre-existing tips; and (c) secondary cracks propagated very quickly and coalesced, meanwhile the outer wing cracks extended

3.2.2

Testing Results

The tests were performed with different crack-bridge geometries. The selected crack inclinations (α) were 30 , 45 and 60 , and the bridge inclinations (β) varied from 45 to 120 in steps of 15 . For each crack-bridge geometry, a minimum of three specimens were tested, one with open cracks and two with closed cracks. Different specimens with the same crack-bridge geometry and crack condition (open or closed) had shown a good reproducibility for both failure pattern and failure load.

3.2.2.1

Process of Crack Propagation and Coalescence

Crack initiation, propagation and coalescence were observed during the tests. A typical sequence of the fracturing process is shown in Fig. 3.3.

30

B. Shen et al.

The failure process shown in Fig. 3.3 is only representative when the inclination of the bridge was low (β60 ). The observed failure processes for open and closed cracks and for different crack-bridge geometries are rather different. Some of the most interesting results which will be discussed in more detail below, are summarized as follows: 1. Wing cracks had different paths for open and closed pre-existing cracks. 2. The coalescence mechanism was very different when the bridge inclination varied from a low angle (60 ) to a high angle (90 ) for both open and closed cracks. The secondary cracks were initiated from the pre-existing crack tips or from the bridge center, depending upon the bridge inclination. 3. For closed cracks, when the crack inclination (α) was less than the friction angle, crack propagation and coalescence did not occur. 4. The load at coalescence of closed cracks was higher than that of open cracks.

3.2.2.2

Wing Cracks

All specimens in our tests showed that the wing cracks emanating from pre-existing open cracks had smaller curvature than those propagating from the pre-existing closed cracks. Wing cracks emanating from pre-existing open cracks formed a greater angle between wing crack and pre-existing crack plan than wing crack propagating from closed pre-existing cracks (see Fig. 3.4). The wing cracks initiated from a closed crack took a path close to a straight line almost parallel to the direction of uniaxial load. The wing crack initiation from a pre-existing closed cracks

wing crack

>90º

100 dB (note that a measuring range of set to 40–100 dB, hence magnitude greater than 100 dB was not able to be detected). For Specimen 2–1 with a crack width l ¼ 40 mm, the stress-time-energy curve and the stress-time-magnitude curve are shown in Fig. 3.25b. When the wing tension crack B1 started, the energy value was 17,829 and the stress had a small sudden drop. When the loading time reached 1576 s, stress had a major drop, and the energy value jumped up to 59,535, but no surface crack was observed. When stress reached the peak strength of 7.5 MPa of the specimen, secondary tensile crack B2 appeared, and the AE energy value climbed rapidly to 18,517. For Specimen 3-2 with a crack width l ¼ 50 mm, the stress-time-energy curve and the stress-time-magnitude curve are shown in Fig. 3.25c. When crack C1 initiated, the AE energy was 11,547, and the stress began to fluctuate. With load increasing,

3 Laboratory Studies of 2D and 3D Rock Fracture Propagation Sta IV

Sta III

Sta II

Sta I

16

34528

14

Stress Energy Index

12

55

35000

16

30000

14

Sta III Sta IV

Sta II

Sta I

Crack A2

Stress Amplitude

100

A3

Crack A1 90

12

25000

15000 6

4 50

5000

2 0 0

500

1000

1500

2 0

0 2500

2000

70

6 60

10000

4

8

Amplitude/dB

8

10

Stress/MPa

Stress/MPa

20000

20448

Energy Index

80 10

0

500

1000

Time/s

1500

40 2500

2000

Time/s

(a)

10

Sta III

Sta IV

50000

8

Sta IV

Sta III

CrackC2

10

60000

59535 Stress Energy Index

Sta II

Sta I

Stress Amplitude

8

90

40000

20000

18517

17829 2

Stress/MPa

30000 4

80

Energy Index

Stress/MPa

6

400

800

1200

1600

2000

70 4 60

CrackC1

0 0

6

2

10000

0

50

0

2400

0

400

800

1200

Time/s

Sta II

Sta I

Sta IV

Sta III

8

CrackD2

Stress Amplitude

7

40 2400

CrackD3

Sta IV

CrackD4

100

90

25000

6

6

20911

80

20000

15000

10000

Stress/MPa

14797 11547

3

Energy Index

4

4

70

3 60

Amplitude/dB

5

5

Stress/MPa

2000

Sta III

Sta II

CrackD1

8

30000

Stress Energy Index 26951

7

1600

Time/s

(b) Sta I

100

Amplitude/dB

Sta II

Sta I

2

2

50

5000

1

1

0

0

0 0

400

800

1200

1600

2000

40 0

2400

400

800

1200

1600

2000

2400

Time/s

Time/s

(c)

Fig. 3.25 Acoustic Emission Characteristic Curve. Notes: the monitored acoustic emission energy signals were the relative energy without dimension. (a) Specimen1-1(l ¼ 30 mm), (b) Specimen 2-1 (l ¼ 40 mm), (c) Specimen 3-2 (l ¼ 50 mm)

56

B. Shen et al.

crack C2 and C3 started, causing stress, AE energy and magnitude to change suddenly. When peak stress was applied on the specimen, secondary tension crack C4 started, causing the AE energy and the magnitude to spike to 14,797 and 98 dB respectively. 1. As shown in Fig. 3.25, the compression of the rock-like specimen with edge crack can be divided into 4 stages namely initial compaction stage, elasticity stage, crack extension stage and specimen failure stage as described below. 2. Initial compaction stage (Stage I): as a rock-like material is homogeneous, with few air bubbles and defects and relatively dense structure, the stress-time curve was relatively flat. The pre-existing open crack in the specimen was deforming with stress increase and no new crack started. The acoustic emission signals were mainly caused by the closure of the open crack under pressure, with relatively low energy and magnitude. 3. Elasticity stage (Stage II): at this stage the stress-time curve exhibited a linear growth for a long period of time. Under the initial pressure, the pre-existing crack within the specimen closed, which created stronger contacting forces and higher friction on the crack surfaces. Hence the propagation of the pre-existing crack and development of new crack were constrained. As a result, no new crack started in the specimen, and acoustic emission signals were weak. 4. Crack extension stage (Stage III): when the stress began to have “stepped” changes, new cracks started and extended around the tip of the pre-existing crack, mainly in the form of tensile crack. With load continuously increasing, the number and length of the new cracks kept growing, causing the AE energy and the magnitude to fluctuate suddenly. At this stage the acoustic emission signals became very active. 5. Specimen failure stage (Stage IV): when the stress reached the peak strength, the cracks extended and eventually connected to each other around the pre-existing cracks. At this stage, the AE energy and the magnitude changed sharply. Then the stress curve of the specimen descended rapidly and the specimen is disintegrated.

3.4.4

Implication of the Study Results to Mine Water Inrush

On Monday, March 1, 2010, a large amount of water flooded the Luotuoshan coal mine in the Inner Mongolia Autonomous Region of China when a roadway was excavated in the No.16 coal seam. The roof of No.16 coal seam is sandstone and mudstone, and the floor of that is mudstone and limestone. With the increase of tunnelling distance, the damage range of floor was enlarged and hidden structure developed by mining, as shown in Fig. 3.26a–c. After mining of coal seam, the roof stratum may collapse, bend or sink. The dynamic loading force produced by the movement of the roof stratum is applied on the coal rock body ahead of the mining face, damaging coal rock body and weakening its water resistance. Under the combined effect of the abutment pressure on the top and the confined water within the floor, the pre-existing cracks in the floor

3 Laboratory Studies of 2D and 3D Rock Fracture Propagation Fig. 3.26 The schematic diagram of Luotuoshan coal mine flood

57

(a)

Tunneling face

NO.16 coal seam

Buried structure

Aquifer

Water flowing fractured zone

(b)

Tunneling face

NO.16 coal seam

Aquifer

Buried structure

Water flowing fractured zone

(c)

Tunneling face

Buried structure

Aquifer

NO.16 coal seam

Water flowing fractured zone

58

B. Shen et al.

Roof NO.1 Roof NO.2 Roof NO.3

Roof pressure

Roof NO.4 Roof NO.5 Roof NO.6 Longwall face

Coal seam

Tensile area Floor failure zone

Water inrush channel

Confined water zone Aquifer

Water pressure

Fig. 3.27 Mechanism for inrush on the floor ahead of the coal mining face. Notes: the left figure depicts the seam roof, the damaged part of the floor and the intrusion belt of the confined water, with the blue arrow indicating the flow direction of the confined water and the black bold line indicating the inrush channel. The problem may be simplified into a specimen with an edge crack as shown in the right figure. The oval part on right represents the tensile region of the specimen when it is under pressure, and the pre-existing crack at the center can be deemed as the coal mining face

strata will further fracture and extend, significantly reducing the thickness of the aquitard in the floor and its water resisting capacity. The confined water can then penetrate into the coal rock body ahead of the mining face through the damage zone created by the extension of pre-existing crack (or mining panel) and enter into the mining panel, causing water inrush into the mining face floor. Further, the principle of the Luotuoshan coal mine flood can be shown as Fig. 3.27. In Fig. 3.27, the inrush channel depicted in the figure essentially resembles with the anti-wing tension cracks A2, B1 and C2 as depicted in Fig. 3.24a–c. Hence the water inrush from the floor ahead of the mining face can be explained well with the propagation mechanisms for edge crack under vertical load.

3.4.5

Conclusions

1. When a rock-like specimen with pre-existing edge crack was under uniaxial compression, tensile cracks firstly appeared in at the tip of its pre-existing crack, with an average crack imitation angle of 84.6 . The new cracks propagated rapidly in an angle close to the loading direction to the upper and lower ends of the specimen. As the width of the pre-existing crack increased, the uniaxial compressive strength and the crack initiation strength of specimen decreased

3 Laboratory Studies of 2D and 3D Rock Fracture Propagation

59

gradually. When the load reached the specimen’s peak strength, anti-wing and secondary tension cracks started and crack extended into the crack-free region and then connected to the pre-existing crack, resulting in tensile fracture in “J” shape or “reverse J” shape and causing the specimen to fail in tension. 2. At the initial stage of loading, the acoustic emission signals from the specimen were weak, with lower AE energy value and magnitude, suggesting that there was no or only minor crack occurring. When the stress changed in “stepped” manner, the specimen produced stronger acoustic emission signals with rapidly increasing energy value and magnitude, indicating the development of macro cracks. 3. The propagation mechanism of pre-existing edge crack under pressure can well explain the inrush on the floor ahead of the mining face. By comparing the propagation path of the anti-wing tensile crack and the path of the inrush on the floor ahead of the mining face, it can be found that they share a strong similarity as there was tensile region formed within the floor rock under the effect of compressive load, so that the confined water can penetrate into the mining face along the tension damage zone, causing water inrush disaster. Acknowledgements Part of the research described in this chapter was financially supported by the Natural Science Foundation of China (No. 51428401), Taishan Scholar Talent Team Support Plan for Advantaged & Unique Discipline Areas, Shandong Provincial Natural Foundation (No. ZR2018LE008), Doctoral Research Foundation of Liaocheng University.

References ASTM E1106-12 (2017) Standard test method for primary calibration of acoustic emission sensors, ASTM International, West Conshohocken. www.astm.org Bazant ZP, Pffeifer PA (1986) Shear fracture tests of concrete. Mater Struct 19:111–121 Deng Q, Zhang P (1984) Research on the geometry of shear fracture zones. J Geophy Res 89:5699–5710 Dyskin AV, Germanovich LN, Ustinov KB (1999) A 3-D model of wing crack growth and interaction. Eng Fract Mech 63(1):81–110 Dyskin AV, Sahouryeh E, Jewell RJ, Joer H, Ustinov KB (2003) Influence of shape and locations of initial 3-D cracks on their growth in uniaxial compression. Eng Fract Mech 70(15):2115–2136 Einstein HH, Veneziano D, Baecher GB, O’reilly KJ (1983) The effect of discontinuity persistence on rock stability. Int J Rock Mech Min Sci Geomech Abstr 20(5):227–237 Griffith AA (1921) The phenomena of rupture and flow in solid. Phil Trans R Soc A A221:163–197 Guo Y, Lin C, Zhu W et al (2008) Experimental research on propagation and coalescence process of three-dimensional flaw-sets. Chin J Rock Mech Eng 27(Supp1):3191–3195 Horri H, Nemat-Nasser S (1986) Brittle failure in compression: splitting, faulting and brittle-ductile transition. Phil Trans R Soc A 319(1549):337–374 Huang M, Wong RHC (2007) Experimental study on propagation and coalescence mechanisms of 3D surface cracks. Chin J Rock Mech Eng 26(9):1794–1799 Jung SJ, Enbaya M, Whyatt JK (1992) The study of fracture of brittle rock under pure shear loading. In: Proceedings of the fractured and jointed rock masses, pp 457–463 Kachanov ML (1982) A microcrack model of rock inelasticity—Part I and Part II. Mech Mater 1: 29–41

60

B. Shen et al.

Kemeny JM, Cook NGW (1986) Effective moduli, non-linear deformation, and strength of a cracked elastic solid. Int J Rock Mech Min Sci 23:107–118 Ketcham RA, Carlson WD (2001) Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences. Comput Geosci 27:381–400 Lajtai EZ (1969) Shear strength of weakness planes in rock. Int J Rock Mech Min Sci Geomech Abstr 6:299–515 Li T, Lu H (2012) CT real-time scanning tests on single crack propagation under triaxial compression. Chin J Rock Mech Eng 29(2):289–296 Li L, Wong RHC, Han Z et al (2012) Experimental and theoretical analyses of three-dimensional surface crack propagation. Chin J Rock Mech Eng 31(2):311–318 Lin P, Zhou Y, Li Z et al (2008) Experimental study on failure behavior on rock containing threedimensional single pre-existing flaw. Chin J Rock Mech Eng 27(Supp2):3882–3887 Melin S (1989) Why are crack paths in concrete and mortar different from those in pmma? Mater Struct 22(1):23–27 Nelson R (1968) Modelling a jointed rock mass. S.M thesis, Massachusetts Institute of Technology Reyes O (1991) Experimental study and analytical modelling of compressive fracture in brittle materials. Ph.D. Thesis of Massachusetts Institute of Technology. Reyes O, Einstein HH (1991) Failure mechanisms of fractured rock: a fracture coalescence model. Proc. 7th Int. Con. on Rock Mechanics, 1, 333–340 Segall P, Pollard D (1980) Mechanics of discontinuous faults. J Geophy Res 85(B8):4337–4350 Shen B (1993) Mechanics of fractures and intervening bridges in hard rocks. PhD dissertation, Royal Institute of Technology, Stockholm Shen B, Stephansson O (1990) Cyclic loading characteristics of joints and rock bridges in a jointed rock specimen. In: Barton N, Stephansson O (eds) Proceedings of the international symposium rock joints, pp 725–729 Shen B, Stephansson O (1992) Deformation and propagation of finite joints in Rock masses. In: Proceedings of the fractured and jointed rock masses, pp 303–309 Shen B, Stephansson O, Einstein HH, Ghahremanl B (1995) Coalescence of fractures under shear stresses in experiments. J Geophys Res 100(B4):5975–5990 Sun X, Shen B, Zhang B (2018) Experimental study on propagation behavior of three-dimensional cracks influenced by intermediate principal stress. Geomech Eng 14(2):195–202. https://doi. org/10.12989/gae.2018.14.2.000 Wong T-F (1982) Micro-mechanics of faulting in Westerly granite. Int J Rock Mech Min Sci 19:49–64 Wong RHC, Huang ML, Jiao MR, Zhu WS (2004) The mechanisms of crack propagation from surface 3-D fracture under uniaxial compression. Key Eng Mater 261-263(I):1–6

Chapter 4

Laboratory Investigations on the Hydraulic Fracturing of Granite Cores Li Zhuang, Sunggyu Jung, Melvin Diaz, and Kwang Yeom Kim

Abstract This chapter introduces laboratory studies on the hydraulic fracturing of granite cores. Some of the most important factors influencing fracturing are considered, including injection rate, fluid infiltration, fluid viscosity, borehole size, and injection scheme. Results from acoustic emission monitoring help elucidate the fracturing process. Hydraulic fractures of granite samples are observed and analysed at the mineral scale with the aid of X-ray scanning and Computed Tomography. A first attempt to investigate the initiation and propagation of hydraulic fractures in granite drill cores during cyclic injection is presented. Keywords Granite · Hydraulic fracturing · AE monitoring · Fracture propagation · Fluid infiltration · Fluid viscosity · X-ray CT

4.1

Introduction

Hydraulic fracturing is widely used for in situ stress measurement and reservoir engineering. Recent developments of enhanced geothermal systems have brought attention to hydraulic fracturing in hot dry rock. Unlike the sedimentary rock formations found in traditional oil and gas reservoirs, geothermal reservoirs usually occur within crystalline rocks such as granite and gneiss. HDR often has low or near-

L. Zhuang Korea Institute of Civil Engineering and Building Technology, Goyang, Republic of Korea e-mail: [email protected] S. Jung · M. Diaz Korea University of Science and Technology, Daejeon, Republic of Korea e-mail: [email protected]; [email protected] K. Y. Kim (*) Korea Institute of Civil Engineering and Building Technology, Goyang, Republic of Korea Korea University of Science and Technology, Daejeon, Republic of Korea e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Shen et al. (eds.), Modelling Rock Fracturing Processes, https://doi.org/10.1007/978-3-030-35525-8_4

61

62

L. Zhuang et al.

zero permeability, which has to be enhanced by hydraulic stimulation to make the reservoir more permeable and hence accessible. There are two main methods of hydraulic stimulation, hydro-fracturing and hydro-shearing, and the choice between them strongly depends on the geological formation and rock stress conditions (Gischig and Preisig 2015). Hydro-fracturing creates new tensile fractures in the rock mass, while hydro-shearing initiates slip at pre-existing fractures and faults. Regardless of the stimulation method, induced seismicity due to fluid injection must be considered (Zang et al. 2019). The injection scheme can be optimized to enhance permeability while controlling the induced seismicity. Many experimental studies have investigated the hydraulic fracturing behaviour of rocks, including sandstone, shale, granite, and artificial rock. The hydraulic fracturing behaviours of different granites have been reported by Haimson (1975, 1999), Haimson and Lee (1980), Solberg et al. (1980), Haimson and Doe (1983), Haimson and Zhao (1991), Zhao et al. (1996), Ishida et al. (1997, 2000, 2004, 2012), Chen et al. (2015), Zhuang et al. (2019a, b) and Stephansson et al. (2019). Various theoretical models have been presented to predict the breakdown pressure (BP) during hydraulic fracturing, but none of these perform well in explaining the observed breakdown phenomena (Guo et al. 1993). Hydraulic fracturing is a complex process, and the initiation and propagation of fractures in different types of rocks have not been fully elucidated. To understand further the hydraulic fracturing behaviour of granites, a series of tests was performed on granite core samples, and the results were analysed with a focus on the influences of injection rate, fluid infiltration, fluid viscosity, borehole size, and injection scheme. In particular, acoustic emission (AE) monitoring and X-ray computed tomography (CT) scanning were employed to investigate the initiation and propagation of hydraulic fracturing and to characterize the fractures.

4.2 4.2.1

Experimental Setup Hydraulic Fracturing Test Equipment

Figure 4.1 shows the equipment used for hydraulic fracturing tests on core samples. Vertical pressure, confining pressure, and injection pressure are supplied by hydraulic pumps. Fluid is injected from the bottom of the rock sample, and rubber O-rings are installed between the sample and the loading plates to prevent leakage of the injected fluid from the top and bottom of the sample. Recorded data include the displacement of the injection piston (for calculating injection volume), the injection pressure and the vertical and confining pressures. Figure 4.2 shows photographs of the main apparatus, including the monitoring and recording unit, the hydraulic pump, and the hydraulic fracturing equipment with triaxial cells mounted on the workbench of the X-ray chamber. The cells are either

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

63

Fig. 4.1 Schematic diagram of the hydraulic fracturing test system

Fig. 4.2 Hydraulic fracturing test equipment: (a) monitoring and record unit, (b) hydraulic pump, (c) steel triaxial cell and (d) PEEK triaxial cell combining with CT scanning

made in steel or polyether ether ketone (PEEK). The maximum confining and injection pressures for the system are 20 MPa and 35 MPa, respectively. Figure 4.3 shows the sequence for setting up the core sample. The sample rests on a pedestal, and injection fluid is supplied from the bottom. Circular plates with O-rings prevent fluid leakage. Later, a thermo-responsive plastic membrane is used to cover the sample, and additional O-rings are placed. The upper part of the equipment, which is responsible for applying vertical load, is suspended by four steel bars. There is a rotation connection at the bottom of the lower platen, which allows the whole loading system to be installed on the rotating table in the X-ray

64

L. Zhuang et al.

Fig. 4.3 Experimental preparation under zero confining pressure: (a) sample setting, (b) membrane and O-ring installation, (c) vertical loading platen setting Fig. 4.4 Granite quarry in Pocheon area, South Korea

chamber. The whole testing system can rotate up to 720
, allowing the sample to be scanned during testing.

4.2.2

Test Samples

The test samples are Pocheon granite, obtained from a quarry in Pocheon, South Korea (Fig. 4.4). X-ray diffraction analysis shows its main components to be plagioclase (35.9%), quartz (35.7%), microcline (25.8%) and biotite (2.6%). The average measured effective porosity is 0.66%. This granite shows mild but clear anisotropy, with three typical cleavages of rift (R), grain (G) and hardway (H), which have different densities of microcracks (Zhuang et al. 2016a). As shown in Fig. 4.5, the samples are classified by the coring direction: the R, G and H labels refer to specimens cored perpendicular to those cleavage planes. Table 4.1 lists the basic physical and mechanical properties of the granite, considering the anisotropy.

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

65

Fig. 4.5 Granite samples: (a) coring from a granite block, (b) Core samples with different relative positions of cleavages Table 4.1 Main physical and mechanical properties of Pocheon granite Specimen /Plane Rift (R) Grain (G) Hardway (H)

Hydraulic conductivitya (m/s) 1.09E-10 8.34E-11 7.20E-11

Elastic modulusb (GPa) 57.53 55.09 58.47

BTSc (MPa) 6.05 8.20 8.83

UCSb (MPa) 220.01 194.62 192.10

Fracture toughnessb (MPa.m0.5) 0.963 1.211 1.583

Zhuang et al. (2019a) Average of three test values b Average of five test values c Average of 20 test values. Hydraulic conductivity, Brazilian tensile strength (BTS) and tensile fracture toughness were measured along the rift, grain and hardway planes. Elastic modulus and uniaxial compressive strength (UCS) were measured on the R, G and H specimens a

4.3 4.3.1

Testing Conditions Stress Conditions

The external stresses applied to a sample directly influence its breakdown pressure and fracture propagation. The first breakdown pressure model by Hubbert and Willis (1957) is Pb ¼ 3σ Hmin  σ Hmax þ σ T  P0

ð4:1Þ

where σ Hmax and σ Hmin are the major and minor horizontal principle stresses, respectively, σ T is the hydraulic tensile strength, and P0 is the initial pore pressure.

66

L. Zhuang et al.

When the vertical stress is the maximum principal stress, the tensile induced hydraulic fractures are vertical, parallel to the maximum horizontal principal stress. Biaxial hydraulic fracturing tests on cylindrical samples employ equal major and minor horizontal principle stresses, which are equal to the confining pressure σ p. Therefore, Eq. (4.1) becomes Pb ¼ 2σ p þ σ T 2 P0

ð4:2Þ

However, poroelastic effects cause experimental breakdown pressures to follow a linear relationship with confining pressure, meaning the coefficient is ~1.0 instead of the theoretical value of 2.0. This phenomenon has also been reported elsewhere for both granite (Rummel 1987) and other types of rock, including sandstone and marble (Brenne et al. 2013). Haimson and Fairhurst (1967) proposed a breakdown pressure model considering the poroelastic effect, which for biaxial test conditions can be expressed as Pb ¼

2σ p þ σ T
 P0 2  α 12υ 1υ

ð4:3Þ

where α is Biot’s poroelstic parameter and υ is Poisson’s ratio.

4.3.2

Injection Fluid and Injection Control

Laboratory hydraulic fracturing experiments can employ a variety of fluids (e.g., water, oil and gases), although water is often used. The viscosity of the injected fluid has a strong influence on the fracturing results (Ishida et al. 2004; Chen et al. 2015), and this will be discussed in the next section. In the laboratory, fluid can be injected by controlling the injection rate or the pressurization rate (Detournay and Cheng 1992). In the field, stimulation is usually controlled by the flow rate. The terms injection rate, flow rate and pumping rate often appear in the literature to represent the amount of fluid injected into a borehole in a certain time.

4.3.3

Hydraulic Fracturing and Sleeve Fracturing

For special tests such as sleeve fracturing, the borehole is covered by a membrane or rubber tube to prevent injection fluid from infiltrating the rock sample, meaning that the poroelastic effect due to fluid infiltration can be ignored.

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

67

Fig. 4.6 Different schemes of injection rate (vertical axis) – time (horizontal axis): (a) Constant rate monotonic injection, (b) Stepwise rate monotonic injection, (c) Cyclic injection and (d) Cyclic progressive injection

4.3.4

Continuous Injection and Cyclic Injection

Conventional hydraulic fracturing usually involves continuous fluid injection until hydraulic breakdown. A massive injection without controlling the injection rate or volume could induce significant seismicity with negative environment effects. Cyclic hydraulic fracturing was proposed by Zang et al. (2013) in an effort to reduce the induced seismicity. It involves alternating between high and low (or zero) injection rates instead of continuous injection, and the injection pressure is lowered frequently to allow stress relaxation at the fracture tip. Figure 4.6 compares two continuous and two cyclic injection schemes, where injection rate varies with the injection time. Recent works have proved the concept and quantitatively evaluated various injection schemes at different scales (Zhuang et al. 2016b, 2017, 2018a, b, 2019b; Zang et al. 2017, 2019; Diaz et al. 2018a, b; Hofmann et al. 2018a, b; Zimmermann et al. 2019).

4.4 4.4.1

Experimental Results Influence of Confining Pressure

Figure 4.7 shows the relationship between BP and confining pressure for R specimens at an injection rate of 25 mm3/s and for H specimens at an injection rate of 100 mm3/s. The specimens have borehole diameter of 8 mm. In both cases, BP increases linearly with increasing confining pressure. Linear fitting yields a ratio

68

L. Zhuang et al.

Fig. 4.7 Relationship between breakdown pressure and confining pressure: (a) R specimen at q ¼ 25 mm3/s, (b) H specimen at q ¼ 100 mm3/s

between them of 0.89 for R specimens and 1.06 for H specimens. The smaller value for the R specimens is due to the lower injection rate. Previous studies have shown that BP is usually higher at a higher injection rate (Zhuang et al. 2019a). According to the first breakdown model suggested by Hubbert and Willis (1957), if the rock is impermeable, the ratio of BP to confining pressure should be 2.0. However, even for low-permeability granite, water infiltrates the specimen in laboratory tests, leading to a non-negligible poroelastic effect. The model of Haimson and Failhurst (1967) in Eq. (4.3) and extensions of this model including Biot’s poroelastic parameter can be used to explain the result. Biot’s poroelastic parameter varies among rock types.

4.4.2

Borehole Size Effect

Haimson and Zhao (1991) reported that breakdown pressures measured in laboratory tests are essentially unaffected by the borehole diameter if it is at least 20 mm; the results are thus directly applicable to the interpretation of field data. However, experimental results by Morita et al. (1996) show a clear decrease in the breakdown pressure of Berea sandstone when the borehole size increased from 38 to 100 mm. Figure 4.8 shows granite samples (all 50 mm diameter, D, and 100 mm height) with cylindrical cross-sections parallel to the grain or hardway plane (i.e., G or H samples). The borehole diameters, d, are 5, 8, 12 and 16 mm, giving d/D ratios of 1:10, 1:6.3, 1:4.2 and 1:3.6, respectively. The influence of the d/D ratio was investigated using water injected at two different rates (50 and 100 mm3/s). Figure 4.9 compares the induced fractures surrounding the borehole. The X-ray CT images have a resolution of 0.0317 mm/pixel and show only the area near the borehole instead of the full section of the specimen. The fractures are generated mainly along the rift plane, which has the weakest tensile strength. Moreover, the case with the smallest borehole diameter of 5 mm has a clear bi-wing fracture.

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

69

Fig. 4.8 Granite samples with different borehole diameters. (Zhuang et al. 2018c)

Fig. 4.9 Induced fractures in the four specimens under injection rate of 50 mm3/s: (a) d ¼ 5 mm, (b) d ¼ 8 mm, (c) d ¼ 12 mm, (d) d ¼ 14 mm. (Zhuang et al. 2018c)

Fig. 4.10 Breakdown pressures at different borehole sizes: (a) Injection rate 50 mm3/s and (b) Injection rate 100 mm3/s. (Zhuang et al. 2018c)

Figure 4.10 compares the average BP and corresponding standard deviations for the four different borehole diameters under injection rates of 50 and 100 mm3/s. In both cases, BP decreases with increasing borehole diameter. For a given borehole diameter, BP is higher at the higher injection rate. The estimated pressurization rate, defined as the slope of the approximately linear section of the injection pressure–time curve before breakdown, is plotted in Fig. 4.11. For a given injection rate, the pressurization rate decreases with increasing borehole size. The magnitude is approximately doubled when the injection rate increases from 50 to 100 mm3/s.

70

L. Zhuang et al.

Fig. 4.11 Comparison of the estimated pressurization rate. (Zhuang et al. 2018c)

4.4.3

Influence of Injection Rate and Fluid Infiltration

Most laboratory hydraulic fracturing tests are performed under injection rate control, largely because fluid injection in the field is usually controlled by injection rate or injection volume. The injection rate and pressurization rate correlate to each other, although the relationship between them may not always be linear. In this section, injection rate influence is investigated with and without considering fluid infiltration. All the specimens have borehole diameter of 8 mm.

4.4.3.1

Injection Rate Effect Considering Fluid Infiltration

Figure 4.12 shows injection pressure with respect to total injected volume for H specimens under seven different injection rates from 1 to 100 mm3/s. The volume is calculated as the injection rate (q) multiplied by injection time (t). The x-axis gives the total injected volume instead of injection time, because the range of the total injection time values is very large, making it difficult to view the result clearly on a linear time scale. Note that the total injected volume does not mean the volume of water infiltrating into the specimen, as some water was discharged through the outer surface of the cylindrical rock sample. The onset section of each curve shows a slow increase in injection pressure, because there was a small volume of air remaining in the booster, which delayed the water compression that provided pressure. The figure shows two curve patterns. The first is for low injection rates (1–25 mm3/s) and has injection pressure increasing slowly with increasing injected volume before converging to a certain value, called the maximum injection pressure Pmax. A peak injection pressure was not observed, indicating a lack of hydrofracturing. The second pattern occurred at high injection rates (50–100 mm3/ s): injection pressure increased more quickly than in the low-injection-rate patterns and reached the breakdown pressure (Pb). After the peak, the zero confining pressure

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

71

Fig. 4.12 Injection pressure plotted with respect to total pumped volume at different injection rates for H specimens at zero confining pressure. (Zhuang et al. 2019a)

Fig. 4.13 Injection pressure and degree of saturation with respect to injection time: (a) Injection rate q ¼ 1 mm3/s and (b) q ¼ 100 mm3/s. (Zhuang et al. 2019a)

caused the injection pressure to drop very quickly to zero. For both types of curve, the ascending section was not linear, especially when approaching Pmax or Pb. Figure 4.13 compares the variations in injection pressure and degree of saturation at a low (1 mm3/s) and a high (100 mm3/s) injection rate. Zhuang et al. (2019a) reported details on the measurement of the degree of saturation. At the low injection rate, the specimen reached full saturation after 2000 s, while the injection pressure was only 0.45 MPa. At saturation, inflow and outflow reached equilibrium, and injection pressure could not increase any more. As a result, the specimen was not hydrofractured. At the high injection rate, the degree of saturation was less than 80% when the injection pressure was close to the BP, meaning that the specimen was fractured before it was fully saturated. Similar to the low injection rate, tests at injection rates of 10 and 25 mm3/s had maximum injection pressures of only 2.06 and 5.06 MPa, respectively, as full saturation was approached. These values are much lower than the threshold BP (~6.0 MPa), and thus the specimens were not hydrofractured. At an injection rate of 50 mm3/s, breakdown was achieved at a degree of saturation of ~80%.

72

L. Zhuang et al.

Numerical simulation on the above experiemental results using UDEC shows that the diffusion length decreases with increasing pressurization rate, as a result breakdown pressure increases (Xie et al. 2018).

4.4.3.2

Injection Rate Effect Excluding the Influence of Fluid Infiltration: Sleeve Fracturing

In sleeve fracturing tests, the borehole and the top and bottom surfaces of the sample were sleeved with a latex cover to prevent water infiltration. Figure 4.14 compares the curves of injection pressure with respect to the total injected volume at injection rates of 1, 10, 50 and 100 mm3/s for tests with and without sleeves. Breakdown was observed in the sleeve fracturing tests at both low and high injection rates. After the early stage, injection pressure increased almost linearly until breakdown, regardless of injection rate. Moreover, injection pressure developed much more quickly in the sleeve fracturing tests, because no water was lost during injection. Figure 4.15 compares the BPs in sleeve fracturing and hydraulic fracturing tests at different injection rates. The sleeve fracturing BPs of H specimens are 18–23 MPa, more than double those in hydraulic fracturing. Established breakdown prediction models that mainly consider in situ stresses, tensile strength and the pore pressure effect are not able to predict the anomalously high BP. This phenomenon was also reported by Brenne et al. (2013), who used a fracture mechanics model to explain the high breakdown pressures in sleeve fracturing tests. Figure 4.16 shows a typical example of injection pressure variation together with the amplitude of induced AE hits detected during the hydraulic fracturing of an H specimen at an injection rate of 100 mm3/s and zero confining pressure. The induced AE starts at an injection pressure of ~7.0 MPa, which is close to the BTS of the specimen’s rift plane. This indicates that cracks were generated near the borehole along the rift plane when the injection pressure exceeded the tensile strength. Fig. 4.14 Comparison of injection pressure variation between tests with and without sleeve in the borehole. (Zhuang et al. 2019a)

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

73

Fig. 4.15 Comparison of breakdown pressures between hydraulic fracturing and sleeve fracturing. (Zhuang et al. 2019a)

Fig. 4.16 Acoustic emission measurements in sleeve fracturing test at q ¼ 100 mm3/s. (Zhuang et al. 2019)

However, with a sleeve around the borehole, there was no water leakage into the specimen, and fluid pressure was not able to reach the inside of the initial cracks to promote fracture propagation. Injection pressure therefore continued to increase linearly because of the confined space of the borehole until the specimen was totally fractured or failed. The resulting breakdown pressure is influenced by the stress gradient in the sample, which depends on the specimen size.

4.4.4

Influence of Injection Fluid Viscosity on Hydraulic Fracturing

Figure 4.17 compares the breakdown pressures measured using tap water and oils injected at different rates. The water has a viscosity of ~1 cp at room temperature,

74

L. Zhuang et al.

while the three tested types of oil have viscosities of 80, 122 and 152 cp and densities of 0.855, 0.858 and 0.861 g/cm3 at 15
C, respectively. Specimens were not fractured by water at low injection rates of 5 and 25 mm3/s, as explained in the previous section. Except for the tests at the very low injection rate of 5 mm3/s, BP increases with increasing viscosity. The volume of infiltrated fluid was measured by weighing samples before and after injection. The results in Fig. 4.18 show that for a given injection rate, the total volume of infiltrated fluid increases with the decreasing viscosity of fluid. The result for the infiltration of oil (122 cp) was excluded as the data were unavailable. Fig. 4.17 Comparison of breakdown pressures using different viscosities of injection fluids

Fig. 4.18 Comparison of infiltrated fluid at different injection rates using water and oils

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

4.4.5

75

Cyclic Hydraulic Fracturing

Laboratory tests of G granite specimens (sample diameter, 50 mm; borehole diameter, 8 mm) were conducted by applying both continuous and cyclic injection schemes. Figure 4.19 compares typical injection pressures, injection rate–time curves and amplitudes of AE hits for both injection patterns. The continuous injection used water at a constant injection rate of 50 mm3/s until breakdown occurred at an injection pressure of 7.0 MPa. Nearly all AE events were detected immediately after the breakdown, and the maximum amplitude was 94 dB. Cyclic injection alternated between a high (50 mm3/s) and zero injection rate, and the interval between them was controlled by the upper and lower bounds of injection pressures for each cycle, which were ~6.2 and ~1.0 MPa in this case. The sample finally failed after 116 injection cycles. The maximum AE amplitude was 72 dB, which was 22 dB lower than that during continuous injection. Injection pressure did not decrease to almost zero like in continuous injection, indicating that the fracture did not propagate to the outside boundary (Zhuang et al. 2019a). Water was still being injected, and the pressure reached a relatively stable value above 4 MPa, which was lower than the given upper bound pressure of 6.2 MPa. No shut-in was performed, and no other cycle was registered.

Fig. 4.19 Injection pressure-injection time curve for (a) monotonic injection and (b) cyclic injection of G specimens

76

L. Zhuang et al.

Fig. 4.20 Breakdown pressure and maximum amplitude of AE event during 5 continuous and 20 cyclic injection tests of granite core samples. Dashed lines indicate average values of monotonic and cyclic injection, respectively. (Zang et al. 2019)

Figure 4.20 shows the maximum amplitude with respect to breakdown pressure for tests conducted under continuous and cyclic injection. Five specimens were tested under continuous water injection (circles) at a rate of 50 mm3/s, and the average breakdown pressure was 7.18 MPa. For the second set of tests, 20 specimens were tested under cyclic injection (triangles in Fig. 4.20). Each test had a maximum injection pressure (lower than the monotonic average) set as an upper limit, and cycling was repeated indefinitely until the specimen failed. These maximum pressures represent a range of 77% to 101% of the average BP measured from hydraulic fracturing tests with continuous injection. The average BP of the 20 cyclic cases was 6.52 MPa, although the specimens failed after different numbers of cycles. The average maximum amplitude of AEs was 93.8 dB for monotonic injection and 67.8 dB for cyclic injection. On average for Pocheon granite, a cyclic injection pattern reduces BP by ~10% and the average maximum magnitude of AE by 26 dB when the experimental setup and samples are otherwise identical.

4.5

Acoustic Emission Monitoring of the Initiation and Propagation of Hydraulic Fractures

AE monitoring is a non-destructive technique used to detect, record or examine acoustic waves generated by diverse sources due to stress changes in a material or structure. Therefore, it is a useful technique for monitoring the structures or crack initiation and propagation in a material subjected to a load. Figure 4.21 shows AE measurements for a cylindrical rock sample loaded vertically. Atomic-scale movements inside the sample generate acoustic waves that are recorded by an AE sensor

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

77

𝜎𝜈

Fig. 4.21 Schematic diagram of AE monitoring during rock sample testing under axial load. (Diaz et al. 2018b)

(25 Mpa) Pre-amplifier

Data Processing 3 2

AE sensors

1

Fig. 4.22 Schematic of a typical AE waveform recorded during monitoring. (MISTRAS 2014)

directly connected to a pre-amplifier signal unit, which sends the data to a computer for storage and analysis.

4.5.1

Basic AE Signal Parameters

A waveform can cause AEs with various properties, although there are some basic parameters used for monitoring and analysis. Figure 4.22 illustrates a typical AE waveform recorded during material testing. The Amplitude represents the maximum positive or negative value during an AE hit; it is typically given in dB. Equation (4.4) describes the relation that defines the amplitude:

78

L. Zhuang et al.

 dB ¼ 20 log

 V max  ðPreamplifier Gain 2 dBÞ 1μ  volt

ð4:4Þ

where Vmax is the maximum amplitude measured in volt and divided by the reference voltage. The preamplifier gain (given in dB) is defined by the user based on the AE source type. Values below a certain level are counted as noise and excluded from the analysis. This Threshold is a crucial parameter that controls the other parameters’ values. Another important parameter is the AE Duration, which is defined as the time between the first and last waveforms above the threshold. Similarly, the Rise time is also affected by the threshold. It is defined as the time from the first waveform above the threshold to the peak amplitude. Finally, the Counts is another value determined from the waveform and defined by the threshold. It refers to the number of AE waveforms that occur above the threshold value. Other derived parameters such as the Average frequency, Energy and Absolute energy are also important. The average frequency is the ratio between the number of AE counts and the AE duration. The energy is calculated as the integral of the voltage signal over the AE hit duration. The absolute energy refers to the true energy of the hit, computed from the integral of the squared voltage signal divided by the reference resistance over the duration, and commonly given in joule.

4.5.2

AE Monitoring Setup

Figure 4.23 depicts the AE monitoring setup during the laboratory hydraulic fracturing tests of granite cores under zero confining pressure (Diaz et al. 2018b). The granite cores are G specimens of the type shown in Fig. 4.5, having a borehole diameter of 5 mm. After the sample is set on the pedestal, an upper cylindrical cap is placed on its top, and the plastic membrane is adjusted. The upper part of the equipment is then carefully placed and rested on four bars with stops. The sample is next loaded to a predefined level before AE sensors are attached to its surface with a coupling agent to facilitate contact between the material and the sensor surface. Confining pressure is not applied. This arrangement allows for AE sensors to be installed directly on the sample (with membrane) to enable the location of AE events during testing. The AE monitoring system (Physical Acoustic Corporation, MISTRAS Group, USA) includes an array of six Nano30 sensors with a frequency domain of 125–750 kHz placed at different heights, as illustrated in Fig. 4.23. The location of each sensor is later represented in the acquisition software to enable event location during post-processing. A pre-amplifier gain value of 40 dB is selected for all the tests.

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

79

Fig. 4.23 Installation of AE sensors on a cylindrical sample

4.5.3

AE Raw Data Filtering

During AE monitoring, many signals are recorded, including noise that should be discarded. Sagar and Prasad (2011) state that low-magnitude signals should be counted as background noise if their duration is less than 10 μs and count less than two. Signals with a duration longer than a pre-defined frame time should also be removed. MISTRAS Group (2014) suggested to use the following equation to define the appropriate hit definition time (HDT): HDT ¼ 1024x

  L  P, S

ð4:5Þ

where L is the signal length in μs, S is the sample rate in millions of samples per second, and P is the pre-trigger time in μs. Another parameter that can be used to discriminate signals is the Energy: signals with an energy equal to zero are advised to be removed (Nguye-Tat et al. 2017).

4.5.4

Analysis Results

The following AE monitoring results were obtained for similar specimens tested under continuous and cyclic injection (as described in Sect. 4.4.5). Figure 4.24 shows the injection pressure, accumulated hits and hit rate with respect to injection time for continuous and cyclic injection. During continuous injection, AE activity was detected immediately before breakdown, at which point the accumulated number of AE hits rapidly increased. Similarly, the hit rate shows a small increment

80

L. Zhuang et al.

Fig. 4.24 Injection pressure, accumulated AE hits and AE hit rate for (a) continuous and (b) cyclic injection of the G specimens

before breakdown followed by the peak value right at the moment of failure. Relatively high hit rates continue for a few seconds as the fracture reaches its final size before the hit rate decreases considerably. On the other hand, the specimen tested under cyclic injection failed after 116 cycles. The accumulated number of hits register early AE activity during the first 4000 s. The extended period of injection, compared with continuous injection, could allow the dilation of pre-existing microcracks in the granite that cause the early AE. The early activity is followed by a period of relatively stable behaviour with minimal AE activity. Close to breakdown, there are two jumps in the accumulated number of hits: one at the last complete cycle and another during the pressure increase of the next cycle that did not reach a pressure above 5 MPa. This implies that the failure occurred in two stages: one small opening during the last complete cycle, followed by the major fracture opening as indicated by the maximum hit rate of 266 hits/s.

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

81

Fig. 4.25 Location of AE events in 2D and 3D for (a) continuous and (b) cyclic injection of the G specimens

Figure 4.25 shows the locations of the AE events for continuous and cyclic injection. Fewer events are plotted for continuous injection, and the AE cloud does not correlate with the fracture size or location. In contrast, the AE events for cyclic injection show a clear orientation along the x-axis (which represents the direction of the rift plane) and most of the AE cloud is located in the lower half of the sample.

4.6

X-Ray CT Observation and Characterization of Hydraulic Fractures

The characterization of length, aperture, roughness and orientation of hydraulic fractures is important, because these properties determine whether hydraulic stimulation has improved reservoir performance. X-ray CT is used here to capture hydraulic fractures, focusing on the initiation and propagation of the fractures during cyclic injection.

82

4.6.1

L. Zhuang et al.

X-Ray CT Equipment

The X-ray CT scanning system (the PCT X-EYE system at the Korea Institute of Civil Engineering and Building Technology, KICT) consists of three different X-ray source tubes with different energy levels (120, 225, and 320 kV) and a flat panel detector of 1024  1024 pixels. The 225 kV source tube for X-ray generation was used to fulfil the optimal scanning condition for the present study, considering the density and size of the specimens. Each specimen was irradiated with X-rays over its circumference by rotating it 360
in 1800 equally spaced increments. Conventionally, artifacts can arise in the reconstructed volume of CT images (Cnudde and Boone 2013). They are caused mainly by the different lengths of the X-ray paths when scanning irregularly shaped materials, and therefore can be largely avoided by using cylindrical specimens. However, ring artefacts can be still found in the images, because they are caused by miscalibration or failure of one or more detector elements in the CT scanner. They occur close to the isocentre of the scan and are usually visible on multiple slices at the same location. This is a common problem in cranial CT and is encountered in this study. Ring artifacts are removed using fast Fourier transform filtering.

4.6.2

Comparison of Hydraulic Fracture Patterns

Figure 4.26 compares representative CT image sections for G specimens fractured in three different cases: monotonic, 473 cycles, and 839 cycles of injection. Zang et al. (2019) reported full details of the test conditions. Quantitative analysis of the fracture paths was possible through extracting patterns from the CT images (Fig. 4.26b), which revealed that monotonic injection led to larger apertures than cyclic fracturing. The complexity of fracture patterns was assessed by computing the tortuosity (length of fracture path divided by the shortest distance between fracture tips), whose values are 1.04, 1.05, and 1.09 for the three cases displayed in Fig. 4.26b, from left to right, although a limited domain is considered for this measurement.

4.6.3

Evolution of Hydraulic Fractures during Cyclic Injection

The evolution of fracturing during cyclic hydraulic injection is monitored here via the first attempt at combining hydraulic fracturing tests (G specimens of Pocheon granite, 30 mm diameter) and CT scanning. The tests were performed outside of the X-ray chamber. Water was injected at a rate of 50 mm3/s up to the point when the injection pressure reached a threshold value of 4 MPa. This value was set by referring to the

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

83

Fig. 4.26 Visualized fracture pattern of granite cores after monotonic and cyclic injection. (a) X-ray CT images and (b) fracture images. (Zang et al. 2019)

breakdown pressure (~6 MPa) during continuous injection at the same injection rate and stress conditions (Zhuang et al. 2018d). Injection was frequently stopped and resumed after the injection pressure was reduced to ~1 MPa to start a new cycle. During testing, the sample was repeatedly CT scanned after the completion of a number of injection cycles (i.e., it was in a stress-free state). The sample finally failed after 29 cycles of injection. Figure 4.27 shows CT images taken at the beginning and after 20, 25, 28 and 29 injection cycles. The images compare the same location, at a height of 1.4 mm from the bottom. Figure 4.27a shows the initial state of the sample before injection; one short pre-existing crack was located near the surface. After 20 cycles (Fig. 4.27b), fractures of different lengths had initiated at both sides of the borehole cutting across quartz and feldspar grains. After five more cycles (Fig. 4.27c), the upper fracture had extended farther, whereas the lower fracture remained unchanged. Later, after three further cycles (Fig. 4.27d), the upper fracture was almost unchanged, while the lower fracture extended to reach the boundary of the feldspar grain. Finally, in Fig. 4.27e, the upper fracture extended along two mica grains, and the lower fracture extended farther along feldspar grain boundaries to the outer surface. The above fracture evolution is limited to the specific examined height of the sample, and different patterns will emerge at different locations. Nevertheless, the evolution indicates that cyclic injection with an injection pressure lower than the monotonic breakdown pressure can propagate fractures, and in particular generate fractures along feldspar–quartz or mica–quartz grain boundaries, which are less strongly bonded than the grains themselves (Zhuang et al. 2018a).

84

L. Zhuang et al.

Fig. 4.27 X-ray CT images and corresponding sketches showing fracture development under cyclic injection. (a) initial state, (b) 20 cycles, (c) 25 cycles, (d) 28 cycles, and (e) 29 cycles. For the sketches, Red line: hydraulic fracture, grey: feldspar grain, black: mica grain. Values represent the total length of fracture at each side of the borehole

4.7

Discussion

The prediction of fracture initiation and propagation in rocks is difficult, particularly for samples with pre-existing cracks. AE monitoring can help elucidate fracture initiation and propagation, but is influenced by many factors, particularly the installation of AE sensors and data filtering. Most BP models work only for specific conditions, as BP is not an inherent characteristic of rock, but depends on various factors including those discussed in this study. This study’s experimental findings on the influence of injection rate, borehole size and viscosity of injection fluid are consistent with past studies on the hydraulic fracturing of granites (Haimson and Zhao 1991; Chen et al. 2015). We noted large differences in the BPs of granite cores fractured hydraulically and using a sleeve, varying with borehole size (Zhuang et al. 2018d) and rock type (Brenne et al. 2013). The linear elastic fracture mechanics model, assuming the pre-existence of a symmetrical double crack with half-length a (Rummel 1987; Haimson and Zhao 1991), is used to interpret the differences. The Pocheon granite used here has a measured tensile toughness KIc of 0.963 MPa∙m0.5 for the rift plane, where fractures were mainly generated. Figure 4.28 compares experimental measurements with theoretical predictions of BP assuming a crack length a ¼ 3.0 mm. Haimson and Zhao (1991) assumed a ¼ 4.3 mm for a different Lac du Bonnet granite to that of the present study. The experimental and theoretical values show similar trends for BP with increasing borehole diameter, despite discrepancies of ~1 MPa for HF tests on large borehole diameters of 12 and 14 mm, and ~2 MPa for all of the sleeve fracturing tests. The fracturing mechanics model performs relatively well in explaining the experimental results.

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

85

Fig. 4.28 Comparison of BP between theoretical and experimental values. HF hydraulic fracturing, SF sleeve fracturing. (Zhuang et al. 2018c)

However, the ideal case of two symmetric pre-existing cracks with the same length near the borehole is rare in the laboratory and practically non-existent in the field. CT observations also show that most hydraulic fractures generated in granite core samples are not symmetric along the borehole axes and that the propagation is not simultaneous even if bi-wing fractures were generated on both sides of borehole. The initiation and propagation of fractures could become more complex when samples become large enough to have significant scale effects. An experimental study on granite blocks by Zhao et al. (1996) showed that the fracture initiation pressure is equal to the breakdown pressure under isotropic far-field horizontal stresses, while for unequal far-field horizontal stresses the fracture initiation occurs considerably before breakdown. The injection fluid (i.e., gas) and rock texture (particularly grain size and pre-existing microcracks) significantly affect the fracturing mechanism of granite (Ishida et al. 2000; Chen et al. 2015). The influence of existing natural fractures was not considered in either above study or this study. The experimental results of this study are limited to uniaxial and biaxial stress conditions, while in situ stresses are usually different for the three principal directions. Further experimental studies on the hydraulic fracturing behaviour of the same Pocheon granite under a true triaxial stress state have been reported by Zhuang et al. (2018a, b) and Diaz et al. (2018a, c).

4.8

Summary

This chapter covers a series of laboratory experimental studies on the hydraulic fracturing of granite cores under uniaxial and biaxial stress conditions. The effects of injection rate, fluid infiltration, fluid viscosity, borehole size and injection scheme are analysed. AE monitoring and X-ray CT technology aided the interpretation of the

86

L. Zhuang et al.

results and shed light on the processes of fracture initiation and propagation. Some key points are summarized as follows. 1. Injection pressure increased with increasing injection rate, although two types of increment pattern were observed. In the hydraulic fracturing tests with fluid infiltration, low injection rates produced slowly increasing pressure curves that reached a certain value with no peak pressure. Conversely, higher injection rates resulted in rapidly increasing pressure curves that led to breakdown with clear peak pressures. 2. During sleeve fracturing tests that excluded fluid infiltration into the rock specimen, breakdown occurred even at low injection rates, and BP was double that observed during hydraulic fracturing at the same injection rate. 3. Breakdown pressure (BP) increased almost linearly with increasing confining pressure, although the rate of increment was affected by the injection rate. BP increased with increasing viscosity of the injection fluid, and it decreased with increasing borehole diameter. 4. Hydraulic fracturing by cyclic injection reduced both the BP and the maximum amplitude of AE compared with those during continuous injection. 5. Finally, CT scanning enabled observation of the fracturing process at the mineral scale for granite cores and characterization of fracture patterns and complexity (i.e., tortuosity).

References Brenne S, Molenda M, Stöckhert F, Alber M (2013) Hydraulic and sleeve fracturing laboratory experiments on 6 rock types. In: Jeffrey R (ed) Effective and sustainable hydraulic fracturing. IntechOpen, London, pp 425–436 Chen Y, Nagaya Y, Ishida T (2015) Observations of fractures induced by hydraulic fracturing in anisotropic granite. Rock Mech Rock Eng 48:1455–1461 Cnudde V, Boone MN (2013) High-resolution X-ray computed tomography in geosciences: a review of the current technology and applications. Earth-Sci Rev 123:1–17 Detournay E, Cheng A (1992) Influence of pressurization rate on the magnitude of the breakdown pressure. In: Tillerson JR, Wawersik WR (eds) Rock mechanics. Balkema, Rotterdam, pp 325–333 Diaz MB, Jung SG, Zhuang L, Kim KY (2018a) Comparison of acoustic emission activity in conventional and cyclic hydraulic fracturing in cubic granite samples under tri-axial stress state. In: Proceedings of the 52nd US rock mechanics/geomechanics symposium. ARMA, Seattle, pp 18–1160 Diaz MB, Jung SG, Zhuang L, Kim KY, Zimmermann G, Hofmann H, Zang A, Stephansson O, Min KB (2018b) Hydraulic, mechanical and seismic observations during hydraulic fracturing by cyclic injection on Pocheon granite. In: Proceedings of the 10th Asian rock mechanics symposium, Singapore Diaz M, Jung SG, Zhuang L, Kim KY, Hofmann H, Min KB, Zang A, Zimmermann G, Stephansson O, Yoon JS (2018c) Laboratory investigation of hydraulic fracturing of granite under true triaxial stress state using different injection schemes – Part 2. Induced seismicity. In: Proceedings of international conference on coupled processes in fractured geological media: observation, modeling, and application, Nov 12–14, Wuhan

4 Laboratory Investigations on the Hydraulic Fracturing of Granite Cores

87

Gischig V, Preisig G (2015) Hydro-fracturing versus hydro-shearing: a critical assessment of two distinct reservoir stimulation mechanisms. In: Proceedings of the 13th international congress of rock mechanics, ISRM 2015, Montréal, Canada Guo F, Morgenstern NR, Scott J (1993) Interpretation of hydraulic fracturing breakdown pressure. Int J Rock Mech Min Sci Geomech Abstr 30(6):617–626 Haimson BC (1999) Six hydraulic fracturing campaigns at the URL, Manitoba. In: Proceedings of 9th ISRM congress, 25–28 August, Paris Haimson B, Doe TW (1983) State of stress, permeability, and fractures in the Precambrian granite of Northern Illinois. J Geophys Res 88(B9):7355–7372 Haimson B, Fairhurst C (1967) Initiation and extension of hydraulic fractures in rocks. Soc Pet Eng J 7(03):310–318 Haimson B (1975) The state of stress in the earth’s crust. Rev Geophys Space Phys 13(3):350–352 Haimson BC, Lee CF (1980) Hydrofracturing stress determination at Darlington, Ontario. In: Proceedings of 13th Canadian symposium on rock mechanics, Canadian Institute of Mining and Metallurgy, pp 42–50 Haimson BC, Zhao Z (1991) Effect of borehole size and pressurization rate on hydraulic fracturing breakdown pressure. In: Roegiers JC (ed) Rock mechanics as a multidisciplinary science. Balkema, Rotterdam, pp 191–199 Hofmann H, Zimmermann G, Zang A, Yoon JS, Stephansson O, Kim KY, Zhuang L, Diaz M, Min KB (2018a) Comparison of cyclic and constant fluid injection in granitic rock at different scales. In: Proceedings of the 52nd US rock mechanics/geomechanics symposium. ARMA, Seattle, pp 18–691 Hofmann H, Zimmermann G, Zang A, Min KB (2018b) Cyclic soft stimulation (CSS): a new fluid injection protocol and traffic light system to mitigate seismic risks of hydraulic stimulation treatments. Geotherm Energy 6:27 Hubbert KM, Willis DG (1957) Mechanics of hydraulic fracturing. Petrol Trans AIME 210:153–168 Ishida T, Chen Q, Mizuta Y (1997) Effect of injected water on hydraulic fracturing deduced from acoustic emission monitoring. Pure Appl Geophys 150:627–646 Ishida T, Sasaki S, Matsunaga I, Chen Q, Mizuta Y (2000) Effect of grain size in granitic rocks on hydraulic fracturing mechanism. In: Proceedings of sessions of Geo-Denver 2000, trends in rock mechanics, geotechnical special publication no.102, ASCE, pp 128–139 Ishida T, Chen Q, Mizuta Y, Roegiers JC (2004) Influence of fluid viscosity on the hydraulic fracturing mechanism. J Energy Resour Technol 126:190–200 Ishida T, Aoyagi K, Niwa T, Chen Y, Murata S, Chen Q, Nakayama Y (2012) Acoustic emission monitoring of hydraulic fracturing laboratory experiment with supercritical and liquid CO2. Geophys Res Lett 39(16), L16309:1–6 Morita N, Black AD, Fuh GF (1996) Borehole breakdown pressure with drilling fluids–I. Empirical results. Int J Rock Mech Min Sci Geomech Abstracts 33(1):39–51 MISTRAS Group Inc. (2014) Express-8 AE system user’s manual Nguyen-tat T, Ranaivomanana N, Balayssac JP (2017) Identification of shear-induced damage in concrete beams by Acoustic Emission. In: Proceedings of the 2nd International RILEM/COST conference on early age cracking and serviceability in cement-based materials and structures – EAC2, Brussels, Belgium Rummel F (1987) Fracture mechanics approach to hydraulic fracturing stress measurements. In: Atkinson BK (ed) Fracture mechanics of rocks. Academic Press, London, pp 217–239 Sagar RV, Prasad BR (2011) An experimental study on acoustic emission energy as a quantitative measure of size independent specific fracture energy of concrete beams. Constr Build Mater 25 (5):2349–2357 Solberg P, Lockner D, Byerlee JD (1980) Hydraulic fracturing in granite under geothermal conditions. Int J Rock Mech Min Sci Geomech Abstr 17(1):25–33

88

L. Zhuang et al.

Stephansson O, Semikova H, Zimmermann G, Zang A (2019) Laboratory pulse test of hydraulic fracturing on granitic sample cores from Äspö HRL, Sweden. Rock Mech Rock Eng 52:629–633 Xie L, Zhuang L, Kim KY, Min KB (2018) Simulating hydraulic fracturing in low permeable rock with consideration of fluid infiltration into rock matrix. In: The 10th Asian rock mechanics symposium, Oct 29–Nov 3, Singapore Zang A, Yoon JS, Stephansson O, Heidbach O (2013) Fatigue hydraulic fracturing by cyclic reservoir treatment enhances permeability and reduces induced seismicity. Geophys J Int 195:1282–1287 Zang A, Stephansson O, Stenberg L, Plenkers K, Specht S, Milkereit C, Schill E, Kwiatek G, Dresen G, Zimmermann G, Dahm T, Weber M (2017) Hydraulic fracture monitoring in hard rock at 410 m depth with an advanced fluid-injection protocol and extensive sensor array. Geophys J Intl 208:790–813 Zang A, Zimmermann G, Hofmann H, Stephansson O, Min KB, Kim KY (2019) How to reduce fluid-injection-induced seismicity. Rock Mech Rock Eng 52:475–493 Zhao Z, Kim H, Haimson B (1996) Hydraulic fracturing initiation in granite. In: Aubertin, Hassani, Mitri (eds) Rock mechanics. Balkema, Rotterdam, pp 1279–1284 Zhuang L, Diaz MB, Jung SG, Kim KY (2016a) Cleavage dependent indirect tensile strength of Pocheon granite based on experiments and DEM simulation. Tunn Undergr Space 26:316–326 Zhuang L, Kim KY, Jung SG, Diaz M, Min KB, Park S, Zang A, Stephansson O, Zimmermann G, Yoon JS (2016b) Laboratory study on cyclic hydraulic fracturing of Pocheon granite in Korea. In: Proceedings of the 50th US rock mechanics/geomechanics symposium. ARMA, Houston, pp 16–163 Zhuang L, Kim KY, Jung SG, Nam YJ, Min KB, Park S, Zang A, Stephansson O, Zimmermann G, Yoon JS (2017) Laboratory evaluation of induced seismicity reduction and permeability enhancement effects of cyclic hydraulic fracturing. In: Proceedings of the 51st US rock mechanics/geomechanics symposium. ARMA, San Francisco, pp 17–757 Zhuang L, Kim KY, Jung SG, Diaz M, Min KB, Park S, Zang A, Stephansson O, Zimmermann G, Yoon JS (2018a) Cyclic hydraulic fracturing of cubic granite samples under triaxial stress state with acoustic emission, injectivity and fracture measurements. In: Proceedings of the 52nd US rock mechanics/geomechanics symposium. ARMA, Seattle, pp 18–297 Zhuang L, Kim KY, Jung SG, Diaz M, Hofmann H, Min KB, Zang A, Zimmermann G, Stephansson O, Yoon JS (2018b) Laboratory investigation of hydraulic fracturing of granite under true triaxial stress state using different injection schemes – Part 1. Permeability enhancement. In: Proceedings of international conference on coupled processes in fractured geological media: observation, modeling, and application, Nov 12–14, Wuhan Zhuang L, Kim KY, Shin HS, Jung SG, Diaz M (2018c) Experimental investigation of effects of borehole size and pressurization rate on hydraulic fracturing breakdown pressure of granite. In: Proceedings the 10th Asian rock mechanics symposium, Oct 29–Nov 3, Singapore Zhuang L, Kim KY, Yeom S, Jung SG, Diaz M (2018d) Preliminary laboratory study on initiation and propagation of hydraulic fractures in granite using X-ray Computed Tomography. In: Proceedings of international conference on geomechanics, geo-energy and geo-resources (IC3G2018), Sep 22–24, Chengdu Zhuang L, Kim KY, Jung SG, Diaz M, Min KB (2019a) Effect of water infiltration, injection rate and anisotropy on hydraulic fracturing behavior of granite. Rock Mech Rock Eng 52:575–589 Zhuang L, Kim KY, Jung SG, Diaz M, Min KB, Zang A, Stephansson O, Zimmermann G, Yoon JS, Hofmann H (2019b) Cyclic hydraulic fracturing of Pocheon granite cores and its impact on breakdown pressure, acoustic emission amplitudes and injectivity. Int J Rock Mech Min Sci 122:104065 Zimmermann G, Zang A, Stephansson O, Klee G, Semiková H (2019) Permeability enhancement and fracture development of hydraulic in situ experiments in the Äspö Hard Rock Laboratory, Sweden. Rock Mech Rock Eng 52:495–515

Chapter 5

Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and Permeability in Rock Mass at 410 m Depth in Äspö Hard Rock Laboratory, Sweden Arno Zang, Ove Stephansson, and Günter Zimmermann

Abstract An underground experiment in Äspö Hard Rock Laboratory is described to bridge the gap between findings from laboratory hydraulic fracturing tests and wellbore-size fluid stimulation in hard rock. Three different water injection schemes are tested quantifying seismic radiated energy, hydraulic energy pumped into the rock, and resulting permeability of the generated fractures. Six hydraulic fracturing tests are performed from a horizontal borehole 102 mm in diameter and 28 m long at 410 m depth and drilled from an existing tunnel in the direction of minimum horizontal stress. Fracture initiation and propagation are mapped by acoustic emission monitoring and impression packer in a rock mass volume 30
30
30 m in size. In tendency, the fracture breakdown pressure is lower and the number of fluid induced seismicity events is smaller if conventional monotonic hydraulic fracturing is replaced by cyclic, progressive injection and/or pulse pumping schemes. The related permeability of the generated fracture can be increased. The maximum permeability increase results from a combination of cyclic and pulse hydraulic fracturing. Laboratory testing of drill-cores from the long borehole show that dynamic pulsing in combination with progressive cyclic pressurization lower fracture breakdown pressure 10–20%. The interpretation for this result is that during fatigue hydraulic fracturing by cyclic/pulse pumping, a larger process zone develops which is accompanied by many smaller seismic events. Ove Stephansson died before publication of this work was completed.

A. Zang (*) · O. Stephansson Section 2.6 Seismic Hazard and Risk Dynamics, German Research Center for Geosciences GFZ, Potsdam, Germany e-mail: [email protected] G. Zimmermann Section 4.8 Geoenergy, German Research Center for Geosciences GFZ, Potsdam, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Shen et al. (eds.), Modelling Rock Fracturing Processes, https://doi.org/10.1007/978-3-030-35525-8_5

89

90

A. Zang et al.

Keywords Fatigue hydraulic fracturing · Fluid-injection-induced seismicity · Underground test · Granitic rock

5.1

Introduction

The recent grow in energy technologies like shale gas and geothermal, as well as the management of subsurface gas reservoirs has led to increased human interaction with rocks in the Earth’s crust. The key ingredient in the discussion of extraction and storage of energy are subsurface fracture systems, their geometry and stability. The instability of fractures is documented in induced seismicity monitored close to the operation sites. This is because human activity perturbs subsurface stresses by fluidinjection or depletion induced by pore pressure changes, causing fractures to grow, coalesce and slip. A classification of fluid-induced seismicity has been suggested e.g. by Ellsworth (2013) and McGarr et al. (2015) though the seismic radiated energy is only a small fraction of the pumped-in hydraulic energy (e.g. in the hydraulic fracture grow process), injection activities are terminated as a result of the felt induced seismicity, e.g. in geothermal operations Giardini (2009). Zang et al. (2014) give an overview of induced seismicity related to geothermal operations and Grünthal (2014), in particular, analyzes the occurrence of seismic events of economic concern. At a specific site the task is to detect, image and control fractures for mechanical and hydraulic integrity of the reservoir under investigation. As field tests in wellbores are time consuming and costly, we see controlled experiments in underground research facilities and mines as a valuable alternative to seek for optimum energy extraction methods with advanced fluid-injection schemes. The optimization process can involve fracture design with minimum seismic radiated energy and/or fracture design with maximum permeability enhancement. For this, radiated seismic energy and pumped-in hydraulic energy needs to be quantified for various injection schemes applied to target rock in well-known stress conditions.

5.2

Underground Experiment in Granitic Rock

The Äspö Hard Rock Laboratory (HRL) is located in the southeastern part of Sweden about 30 km north of the city of Oskarshamn. It has been selected to fix relevant variables for performing controlled fluid injection experiments. In the first place, the stress state at 410 m depth is fixed, and well known from hydraulic fracturing by Klee and Rummel (2002) and overcoring stress measurements by Ask (2006). Second, the granitic rock types are fixed, and relevant for geothermal reservoir development. Äspö granodiorite, gabbro and fine-gained granite are well characterized, both mechanically and hydraulically. Third, the total volume of fluid injected is limited to 30 liters maximum. This allows investigating the impact of injection style on seismic radiated and hydraulic energy in naturally fractured granitic rock mass with size of about 30
30
30 m, see Fig. 5.1 after Zang

5 Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and. . .

91

Fig. 5.1 Hydraulic fracture design and high-frequency sensor array at 410 m depth in Äspö HRL. (a) Map view of horizontal fracturing borehole F1 drilled in the orientation of minimum horizontal stress. Monitoring boreholes (M1– M3) are equipped with AE sensors. Short holes in the tunnel roof are equipped with both AE sensors and accelerometers. (b) Installing a chain of AE sensors in tunnel TASN

et al. (2017). Six borehole intervals free of visible fractures are identified for hydraulic testing. Three different water-injection schemes are applied (Fig. 5.2). Conventional hydraulic fracturing (HF) with continuous fluid injection follow the ISRM suggested method for hydraulic fracturing stress measurements Haimson and Cornet (2003). The typical test starts with the initiation of the packer system to seal the interval, followed by a rapid pressurization with water to test the system for potential leakage (pulse integrity test). During the subsequent main test phase a constant injection rate is applied (see Fig. 5.2a). Pressure increases until it reaches the fracture breakdown pressure (FBP), followed by a decline to a stable pressure level called the fracture propagation pressure. After stable pressure conditions are reached, the well is shut-in and the pressure drops rapidly to the instantaneous shutin pressure (ISIP) followed by a decline curve. The ISIP is assumed to be equivalent to the minimum principal stress. Finally, the interval pressure is released and the

92

A. Zang et al.

Fig. 5.2 Schematics of the three hydraulic testing methods applied in the horizontal borehole at 410 m depth in Äspö HRL; (a) conventional test with constant flow rate, (b) progressively increasing flow rate with cyclic flow rate, and (c) progressively increasing and pulsed cyclic flow rate. (Modified from Zimmermann et al. 2019)

fluid volume is recovered. The test procedure is repeated several times to obtain the fracture reopening pressure (FRP) at each test cycle. The tensile strength of rock is calculated from FBP-FRP. It is assumed that the fracture has been closed completely in between the cycles. The test procedure for the progressive water injection consists of a modified pressure scheme. First the pressure is increased to 20% of FBP obtained from the conventional test in the same formation. Then, a shut-in for several minutes follows with subsequent pressure release (phase of depressurization). Then, the pressure is increased by a level ca. 10% above the previous pressure level following the same scheme. After shut-in another depressurization phase follows. Sequences of pressurization and depressurization alternate until a pressure drop indicates rock failure. This treatment is best described as a cyclic hydraulic pressure scheme with progressively increasing flow rate, see Fig. 5.2b. The subsequent re-fracturing stages follow the same scheme like in the conventional test. The treatments differ only for pressures below the FBP, i.e. single-flow rate tests are replaced by multiple-flow rate fracture breakdown tests. The fatigue hydraulic fracturing (FHF) test procedure is a combination of the progressive injection test and a pulse hydraulic fracturing (PHF) test. The hydraulic

5 Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and. . .

93

equipment for the pulse dynamic test consists of a hydraulic pump to maintain linear and dynamic pressure levels (pressurization bands) together with a second hydraulic pressure pump to drive the dynamic pulse tool with adjustable amplitude and frequency (5–20 Hz). Both pressure signals are combined to result in dynamic pressure pulses on different predefined hydraulic pressure levels based on the FBP of conventional tests carried out previously. The FHF test is best described as a test with cyclic increasing target pressure and additional high frequency pressure oscillations in pressurization phases with step-wise increasing target pressure and flow rate, see Fig. 5.2c. Also this test procedure is frequently interrupted by depressurization phases where crack tip stresses are released. The in situ testing of different injection schemes with monitoring associated seismic and electro-magnetic signals is conducted in the horizontal borehole F1, 102 mm in diameter and 28 m long, drilled from tunnel TASN in the direction of minimum horizontal stress (Fig. 5.1a). Fluid-induced hydraulic fractures open and propagate in radial planes following the on-site in situ stress conditions (SV¼SH > Sh). The high frequency seismic network used consists of 11 acoustic emission (AE) sensors (Fig. 5.1a, red triangles, frequency 1–100 kHz) and four accelerometers (Fig. 5.1a, orange triangles, frequency below 25 kHz). AE sensors are implemented in three monitoring boreholes located left and right of the hydraulic testing borehole (Fig. 5.1a, M1–M3). In Fig. 5.1b, the borehole implementation of a chain of AE sensors is shown. The remaining AE sensors and accelerometers are implemented in short boreholes in the roof of the surrounding tunnels. This monitoring design allows following hydraulic fracture nucleation and growth since the fracture opens perpendicular to the minimum principal stress Sh, and rapidly grows in the plane containing the intermediate SH and maximum principal stress SV, which is the direction towards the monitoring boreholes (Zang et al. 2017). A continuous and a triggered recording system are in operation at 1 MHz sampling rate. This allows near real-time tracking of the hydraulic fracture growth process Kwiatek et al. (2017), and post processing of full waveforms Lopez Comino et al. (2017) for source analysis. From a seismological point of view, full waveforms recorded during injection tests have excellent data quality for further processing which includes hypocenter locations, relocations, and in depth investigations of fracture source, growth and coalescence.

5.3

Results

In this section, the seismic response of the high-frequency sensor network from six hydraulic tests in the horizontal borehole at 410 m depth is presented with the permeability enhancement process inferred from hydraulic data, and the fracture pattern inferred from impression packer test results. Laboratory tests with cyclic increasing target pressure and additional high frequency pressure oscillations in pressurization phases with step-wise increasing target pressure were conducted on

94

A. Zang et al.

90 mm drill-cores taken at the location of the six test intervals in the horizontal borehole F1.

5.3.1

Seismic Radiated Energy

In Fig. 5.3, hydraulic test parameters like FBP, FRP, flow rate and injected volume are compared with the total number of AE events localized (color bars) for different rock types and injection schemes. In the deeper section of the horizontal borehole, F1 three hydraulic tests are carried out in Ävrö granodiorite, two conventional continuous injection tests and one progressive cyclic injection test. Compared to the conventional tests (HF1, HF2), the AE activity in the progressive test (HF3) starts at a later stage of the treatment, and the total number of seismic events is less. Experiment HF3 reveals a FBP of approximately 9.2 MPa. The conventional test with continuous water injection in the same rock type (HF2), leads to a value of FBP 15% larger than compared to the cyclic, progressive test. The two tests in diorite

Fig. 5.3 Number of localized seismic events is shown per fracturing stage for six hydraulic tests in horizontal borehole F1 at 410 m depth in Äspö HRL. In each experiment, the fracture breakdown pressure, fracture reopening pressure, flow rate range, and total volume of water injected is listed. The first test, HF1 starts in the deeper part of the 28 m long, horizontal borehole (25 m), and the last test HF6, is operated about 5 m from the onset of F1 at the tunnel wall

5 Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and. . .

95

gabbro generated only one AE event indicating that this rock type is less seismic active than the Ävrö granodiorite when monitored with the same trigger level. In test HF5, the progressive pulse testing is applied mimicking the FHF treatment with the high frequency oscillation pressure generated by the dual pump system. This test results in a lower FBP (9.0 MPa) compared to the conventional test in the same rock HF4 (FBP ¼ 10.6 MPa), see Fig. 5.3. In the fine-grained granite close to the tunnel wall (HF6), seismic activity is observed in all fracturing stages. In contrast to other tests, in the granite the maximum number of seismic events occurred during early stages of the hydraulic fracturing and re-fracturing experiment, Fig. 5.3. In Fig. 5.4, the evolution of acoustic emission events over time is shown for neighboring test intervals (HF2 and HF3) with different injection schemes in the same rock, Ävrö granodiorite. For this, the hydraulic data (injection pressure and flow rate) are plotted together with the AE activity. In Fig. 5.4a, the conventional HF test HF2 and in Fig. 5.4b, the progressive cyclic test scheme H3F is shown for a full sequence of fracturing and re-fracturing stages. While AE events are observed during all fracturing stages (except refrac2) of the conventional test HF2 (Fig. 5.4a), in experiment HF3 with cyclic progressive water injection, AE events occur in the third and fourth re-frac stage, only (Fig. 5.4b). No AE occur before the FBP in the second last cyclic, progressive treatment in test HF3 despite the steady increase of flow rate for the last three cycles.

5.3.2

Permeability Enhancement Process

In Fig. 5.5, the permeability enhancement process in Ävrö granodiorite is compared for the two schemes HF2 and HF3 displayed in Fig. 5.4. The permeability is estimated from the hydraulic pressure decay curves, Zimmermann et al. (2019). The initial permeability of rock is estimated from the first data point in the test, e.g. 0.1 mD (Fig. 5.5, HF2). In this test, the increase in permeability is 1.3 mD after the initial fracturing and 4.8 mD after the last refrac stage. In the same period of time, eight AE are registered before the FBP, and totally 102 AE at the end of the experiment. In the cyclic progressive test HF3, seismicity is observed in the last two re-fracturing stages, only (Fig. 5.4b). The permeability, however, increases monotonically in cyclic treatment from 0.2 mD before FBP to 2 mD after the last re-fracturing cycle (refrac4), see Fig. 5.5. This demonstrates that the injection style has a strong impact on both, the total number of seismic events observed (conventional 102, cyclic progressive 16), and the maximum magnitude of the AE events (conventional 49 dB, cyclic progressive 43 dB). The permeability enhancement process shows a higher permeability for conventional HF 5.0 mD and less for cyclic progressive treatment, 2 mD. Note that the rock permeability may increase while the associated fracturing mechanisms are below the AE detection limit. The maximum permeability performance was reached

96

A. Zang et al.

Fig. 5.4 Injection pressure and flow rate (left axis, red and blue) and AE magnitude (right axis, red dots) versus time for hydraulic fractures generated with two different injection schemes: (a) conventional hydraulic fracturing HF2, and (b) cyclic, progressive injection scheme HF3 – one method to mimick fatigue hydraulic fracturing. (Modified after Zang et al. (2017)

5 Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and. . .

97

Fig. 5.5 Permeability inferred from pressure decay curves in Ävrö granodiorite for neighboring test intervals with two different water-injection schemes versus time: conventional hydraulic fracturing HF2, and cyclic, progressive injection scheme in experiment HF3. (Modified after Zimmermann et al. 2019)

by progressive pulse testing (Fig. 5.2c). However, no acoustic emissions could be detected in the corresponding rock type, diorite gabbro.

5.3.3

Fracture Pattern from Impression Packer

When comparing the orientation of hydraulic fractures in Ävrö granodiorite we find from impression packer results a single fracture plane in conventional testing (HF2), and we find two fracture planes in the cyclic, progressive test HF3. This finding is in line with results from the AE hypocenter distributions which indicate a single fracture plane for tests HF2 while in test HF3, a more complex fracture pattern is evident, see Fig. 5.6. The fracturing test HF5 was performed as a dynamic pulse test with progressively increasing flow rate and pressure pulse on top. The test was carried out at a borehole depth of 13.3 m in fine grained diorite-gabbro. During the whole HF5 test no AE events was recorded while permeability increased from 2.3 mD to more than 25 mD. The minor fracture trace recorded from the impression packer is shown in Fig. 5.6. The strike direction and dip angle of the recorded fractures from the impression packer tests and the orientation of the major S1, intermediate S2 and minor principal stress S3 from overcoring stress measurements conducted by Ask (2006) are presented in Fig. 5.7. The majority of induced hydraulic fractures are aligned with the direction of the maximum principal stress and dipping in average 70 degrees in the direction of the intermediate principal stress. Kwiatek et al. (2018) investigated the source characteristics of picoseismicity recorded during the six hydraulic fracturing in situ experiments, HF1-HF6. The combined seismic network allowed for detection and detailed analysis of 196 small scale seismic events with moment magnitudes Mw < 3.5 that occurred during the stimulations and shortly after, see Fig. 5.8.

98

A. Zang et al.

Fig. 5.6 Impression packer test images of hydraulic fracturing experiments in borehole F1; (a) HF2 at 22,5 m depth in Ävrö granodiorite; (b) HF 3 at 19.0 m depth in Ävrö granodiorite and (c) HF5 at 11.3 m in diorite gabbro. (After Zimmermann et al. 2019)

5.4

Discussion

At mine scale with decameter size hydraulic fractures, FBP in Ävrö granodiorite is lowered by 15% when using neighboring injection intervals, and replacing monotonic (10.9 MPa) by cyclic, progressive water injection (9.2 MPa). The total number of AE located is reduced from 102 events (HF2, monotonic injection) to 16 AE events (HF3, cyclic progressive injection). The combination of cyclic and pulse hydraulic fracturing (HF5) resulted in zero AE but in a different rock type (diorite gabbro). Compared to the initial value of permeability in the conventional hydraulic fracturing test HF2, the permeability increases by 3.5 mD compared to 1.8 mD in the cyclic progressive injection case HF3. The trend of permeability increase is less advanced for the cyclic injection scheme compared to the conventional test, but also leads to lower seismicity. In case further experiments confirm these findings, it can be concluded that, to achieve a similar permeability increase, a cyclic treatment needs to have a longer duration and hence will be more costly. With respect to field applications, it seems to be a feasible option to reduce the risk of unwanted seismic

5 Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and. . .

99

Fig. 5.7 Fracture location and orientation from impression packer of all tests in borehole F1. The strike of majority of generated fractures agrees with the azimuth of the maximum principal stress S1 and the dip is sub-parallel with the dip and dip direction of the least principal stress S3 from overcoring data

Fig. 5.8 Spatial view of rock mass from NW. The seismic activity is shown with spheres reflecting the stimulation stage and size corresponding to the moment magnitude. Colored disks reflect the stimulation intervals in borehole F1. From borehole bottom to the onset: green (HF1); blue (HF2); red (HF3); teal (HF4); magenta (HF5) and yellow (HF6). Yellow and green bottle-shaped objects are AE sensors and accelerometers. (After Kwiatek et al. 2018)

100

A. Zang et al.

events and hence foster the public acceptance. In addition, the fracture from impression packer indicates a single fracture plane in conventional tests (HF1). We see acoustic emission events as a result of originating from process zone related micro-shears in Ävrö granodiorite while we agree with Warpinski et al. (2012) that the amount of energy radiated as elastic waves is only a small fraction of the injection energy of the pump. Stephansson et al. (2019) performed laboratory pulse tests in combination with cyclic increasing pressurization on cores from the test intervals of the hydraulic fracturing borehole F1 at Äspö HRL. Two cores from each of the three different rock types are tested and after each cycle a 3-min-long dynamic pressure pulses of amplitude 4 MPa and frequency 1 Hz are superimposed at each progressive cycle with constant pressure. The majority of the laboratory tests of different rock types at Äspö HRL show that dynamic pulsing in combination with progressive cyclic pressurization reduces the breakdown pressure of the order of 10–20% for the fine-grained granite and diorite-gabbro, respectively. No reduction in breakdown pressure from cyclic pressurization and dynamic pulsing is observed for the two core samples of Ävrö granodiorite. The most likely explanation for the lack of reduction is the mineral composition and degree of metamorphose of the granodiorite and a higher hydraulic strength of the sample tested with cyclic increasing pressure and pulses. We see the larger fracture process zone with more complex fracture pattern evolving during cyclic progressive and pulse hydraulic fracturing as a result of frequent changes of the fracture direction caused by depressurization phases and stress relaxation at the fracture tip. Stress changes can be explained by (a) pore pressure increase due to freshly created fractures in the process zone, and by (b) rock chips removed from fracture faces by oscillations during fatigue hydraulic fracturing. As a consequence, larger seismic events in conventional hydraulic fracturing are mitigated by smaller seismic events caused by arresting and branching fractures during the fatigue treatment. In terms of energy balance, less fracture energy is needed and less seismic energy is radiated, if a hydraulic fracture runs through a rock volume which is efficiently fragmented beforehand.

5.5

Conclusion

In this study, advanced fluid injection schemes are tested in well controlled rock and stress conditions at mine scale and in the laboratory. At mine scale, the horizontal borehole for fluid injection at 410 m depth in Äspö HRL has a diameter of 102 mm and a total length of 28 m. Hydraulic tests in naturally fractured granite with maximum 30 liter of water injected generated small-scale hydraulic fractures with extension ca. 20–30 square meter. The seismic response of the hydraulic fracture strongly depends on injection style and rock type. In the same rock, cyclic, progressive injection and pulse injection produced less acoustic emission activity as compared to conventional, continuous fluid-injection. Also, the fracture pattern inferred

5 Impact of Injection Style on the Evolution of Fluid-Induced Seismicity and. . .

101

from impression packer results and acoustic emission hypocenter solutions turns out to be more complex when replacing conventional by fatigue hydraulic fracturing. We see the larger fracture process zone with more complex fracture pattern evolving during cyclic progressive and pulse hydraulic fracturing as a result of depressurization phases and stress relaxation at the fracture tip. This conclusion applies to laboratory and mine scale tests. As a consequence, larger seismic events in conventional hydraulic fracturing are mitigated by smaller seismic events caused by arresting and branching fractures during the fatigue treatment. Acknowledgements The in situ experiment at Äspö Hard Rock Laboratory (HRL) was supported by the GFZ German Research Center for Geosciences (75%), the KIT Karlsruhe Institute of Technology (15%) and the Nova Center for University Studies, Research and Development Oskarshamn (10%). We thank Gerd Klee, MeSy Solexperts and Hana Semikova, ISATech Ltd. for performing the hydraulic testing and Göran Nilsson, GNC for arranging the diamond drillings. We thank Katrin Plenkers and Thomas Fischer (GMuG) for the implemention of acoustic emission sensors and accelerometer. An additional in-kind contribution of the Swedish Nuclear Fuel and Waste Management Co (SKB) for using Äspö HRL as test site for geothermal research is greatly acknowledged. We also acknowledge the technical assistance of O. Vanecek, Industrial Safety Assessment Technique, ISATech, Prague, Czech Republic.

References Ask D (2006) Measurement-related uncertainties in overcoring data at the Äspö HRL, Sweden. Part 2: Biaxial tests of CSIRO HI overcore samples. Int J Rock Mech Min Sci 43(1):127–138 Ellsworth WL (2013) Injection-induced earthquakes. Science 341:1225942. https://doi.org/10. 1126/science.1225942 Giardini D (2009) Geothermal quake risk must be faced. Nature 426:848–849 Grünthal G (2014) Induced seismicity related to geothermal projects versus natural tectonic earthquakes and other types of induced seismicity events in Central Europe. Geothermics 52:22–35 Haimson BC, Cornet F (2003) ISRM suggested methods for rock stress estimation-part 3: hydraulic fracturing (HF) and/or hydraulic testing of pre-existing fractures (HTPF). Int J Rock Mech Min Sci 40(7–8):1011–1020 Klee G, Rummel F (2002) Rock stress measurements at the Äspö HRL Hydraulic fracturing in boreholes, Technical Report IPR-02-02. Stockholm, SKB Kwiatek G, Plenkers K, Martinez Garzon P, Leonhardt M, Zang A, Dresen G (2017) New insights into fracture process through in-situ acoustic emission monitoring during fatigue hydraulic fracture experiment in Äspö Hard Rock Laboratory. Proc Eng 191:618–622 Kwiatek G, Martínez-Garzón P, Plenkers K, Leonhardt M, Zang A, von Specht S et al (2018) Insights into complex sub-decimeter fracturing processes occurring during a water injection experiment at depth in Äspö Hard Rock Laboratory, Sweden. J Geophys Res Solid Earth 123:6616–6635 Lopez Comino J, Heimann S, Cesca S, Grigoli F, Milkereit C, Dahm T, Zang A (2017) Characterization of hydraulic fractures growth during the Äspö Hard Rock Laboratory Experiment (Sweden). Rock Mech Rock Eng 50:2985–3001 McGarr A, Bekins B, Burkardt N, Dewey J, Earle P, Ellsworth W, Ge S, Hickman S, Holland A, Majer E, Rubinstein J, Sheehan A (2015) Coping with earthquakes induced by fluid injection. Science 347(6224):830–831

102

A. Zang et al.

Stephansson O, Semikova H, Zimmermann G, Zang A (2019) Laboratory pulse test of hydraulic fracturing on granitic sample cores from Äspö HRL, Sweden. Rock Mech Rock Eng 52:629–633 Warpinski NR, Du J, Zimmer U (2012) Measurements of hydraulic-fracture-induced seismicity in gas shales. SPE Prod Op 27:240–252 Zang A, Oye V, Jousset P, Deichmann N, Gritto R, McGarr A, Majer E, Bruhn D (2014) Analysis of induced seismicity in geothermal reservoirs – an overview. Geothermics 52:6–21 Zang A, Stephansson O, Stenberg L, Plenkers K, Specht S, Milkereit C, Schill E, Kwiatek G, Dresen G, Zimmermann G, Dahm T, Weber M (2017) Hydraulic fracture monitoring in hard rock at 410 m depth with an advanced fluid-injection protocol and extensive sensor array. Geophys J Int 208(2):790–813 Zimmermann G, Zang A, Stephansson O, Klee G, Semikova H (2019) Permeability enhancement and fracture development of hydraulic fracturing experiments in the Äspö Hard Rock Laboratory, Sweden. Rock Mech Rock Eng 52:495–515

Part II

Numerical Methods

Chapter 6

Modelling Rock Fracturing Processes with FRACOD Baotang Shen, Ove Stephansson, and Mikael Rinne

Abstract Rock failure is often controlled by fracture initiation, propagation and coalescence, especially in hard rocks where explicit fracturing rather than plasticity is the dominant mechanism of failure. Prediction of the explicit fracturing process is therefore necessary when the rock mass stability is investigated for engineering purposes. However, the fracture mechanics approach is rarely used in practical rock engineering design partly due to the inadequate understanding of complex fracturing processes in jointed rock mass, and partly due to the lack of tools which can realistically predict the complex fracturing phenomenon in rock mass. Since the 1990s, a new approach to simulating rock mass failure problems has been developed using a numerical code called FRACOD. FRACOD is a code that predicts the explicit fracturing process in rocks using fracture mechanics principles. Over the past three decades, significant progress has been made in developing this approach to a level that it can predict actual rock mass stability at an engineering scale. The code includes complex coupling processes among the rock mechanical response, thermal process and hydraulic flow, making it possible to handle coupled problems often encountered in geothermal extraction, hydraulic fracturing, nuclear waste disposal, and underground LNG storage. During the last three decades, numerous application cases have been conducted using FRACOD, which include: borehole stability in deep geothermal reservoir, Ove Stephansson died before publication of this work was completed.

B. Shen (*) CSIRO Mineral Resources, Brisbane, Queensland, Australia Shandong University of Science and Technology, Qingdao, China e-mail: [email protected] O. Stephansson GFZ German Research Centre for Geosciences, Potsdam, Brandenburg, Germany Royal Institute of Technology (KTH), Stockholm, Sweden M. Rinne School of Engineering, Department of Civil Engineering, Aalto University, Espoo, Finland e-mail: mikael.rinne@aalto.fi © Springer Nature Switzerland AG 2020 B. Shen et al. (eds.), Modelling Rock Fracturing Processes, https://doi.org/10.1007/978-3-030-35525-8_6

105

106

B. Shen et al.

pillar spalling under mechanical and thermal loading; prediction of tunnel and shaft stability and excavation disturbed zone (EDZ), etc. This Chapter summarizes the theoretical fundamentals of the fracture mechanics approach with FRACOD, and the most recent developments. Several validation cases are included in this chapter to demonstrate the validity of this approach. Keywords Rock mass · Fracture · Propagation · Coupling · FRACOD · Thermal · Fluid flow

6.1

Introduction

Rock fracture mechanics is a promising outgrowth of rock mechanics and fracture mechanics, and it has developed rapidly in recent years, driven by the need for in-depth understanding of rock mass failure processes in both fundamental research and rock engineering designs and construction. As rock engineering extends into more challenging fields (like mining at depth, radioactive waste disposal, geothermal energy, and deep and large underground spaces), it requires knowledge of the rock mass’s complex coupled thermal-hydraulic-chemical-mechanical processes. Rock fracture mechanics plays a crucial role in these complex coupled processes simply because rock fractures are the principal carrier (e.g. fluid flow) and common interface (e.g. heat exchange between rock and fluid). To date, the demand for rock fracture mechanics-based design tools has outstripped the very limited number of numerical tools available. Most of those tools were developed for civil engineering and material sciences and deal with substances such as metals, ceramic, glass, ice, and concrete which differ markedly from rocks in their fracturing behaviour. Since the early 1990s, a new approach has been taken to develop a practical numerical approach using fracture mechanics principles to predict rock mass failure processes. The development was based on several laboratory studies and new understandings about rock fracture propagation and coalescence mechanisms (Reyes and Einstein 1991; Shen et al. 1995), in particular the acceptance of existence of shear fracture (mode II) propagation in rock masses and its critical role in rock mass failure in a compressive stress environment. Led by this understanding, a new fracture criterion was proposed (Shen 1993; Shen and Stephansson 1994) that predicts both tensile and shear fracture propagations, overcoming the shortcomings of traditional fracture criteria that predict only tensile failure. This approach has proved very effective in simulating the behaviour of multiple fractures in rock-like materials in laboratory tests (Shen and Stephansson 1993). Development of this modelling approach with a view to engineering application was initially driven by proposals for radioactive waste disposal in Sweden and Finland, where the activation and propagation of fractures initiated by thermal

6 Modelling Rock Fracturing Processes with FRACOD

107

loadings and glaciations are considered major risk factors (Rinne 2000; Rinne et al. 2003; Shen et al. 2004). During this period, an earlier version of the code FRACOD was developed, capable of simulating fracture propagation, fracture initiation, and acoustic emission. This code capability was then expanded to include time-dependent rock behaviour and subcritical crack growth (Rinne 2008). In the course of this development process, many application case studies were conducted using FRACOD, including the well-known Äspӧ Hard Rock Laboratory’s Pillar Spalling Experiments (APSE) in Sweden (Rinne et al. 2004), the DECOVALEX International Collaboration Project (Rinne and Shen 2007), and the Mizunami Underground Research Laboratory (MIU) Investigations in Japan (Stephansson et al. 2003; Shen et al. 2011). This fracture mechanics approach was further expanded to other application fields of rock engineering such as tunnelling and geothermal energy. In an attempt to investigate the stability of a tunnel under high horizontal stresses, FRACOD successfully predicted the same “log-spiral” type of fracturing pattern around the tunnel that was observed in the laboratory (Barton 2007). When applied to the backanalysis of in-situ stresses in a 4.4 km deep geothermal well in Australia, this approach was shown to realistically simulate the borehole breakout, thereby accurately predicting the rock mass stress state (Shen 2008). The concerns about climate change have significantly increased worldwide interest in alternative energy sources and storage methods. Thus, accurate prediction of the coupled behaviour of rock fracturing, fluid flow and thermal processes is now a vital scientific endeavour. FRACOD seeks to address the complex design issues facing various emerging developments in energy-related industries including geothermal energy by enhanced geothermal system, LNG underground storage and CO2 geosequestration. Since 2007, the focus of FRACOD development has shifted to the coupling between rock fracturing, fluid flow and thermal loading, thanks to the establishment of an international collaboration project with participants from Australia, Europe, USA, South Korea and China. Coupling functions of Mechanical-Thermal-Hydraulic (M-T-H) processes have been developed in FRACOD (Shen et al. 2009, 2012). Several application case studies related to hydraulic fracturing, LNG underground storage and nuclear waste disposal (underground borehole spalling experiments) in Finland have been conducted (Xie et al. 2014; Shen et al. 2008; Siren et al. 2014). Development and application of the fracture mechanics approach are also being extended to a three-dimensional version of FRACOD for modelling true 3D problems (Shi and Shen 2014; Shi et al. 2014). This chapter summarises the key physical and theoretical foundation behind FRACOD and describes several application cases solving actual industry problems, with a focus on hydraulic fracture modelling. For those who wish to read more details about FRACOD, please refer to the book “Modelling Rock Fracturing Processes” by Shen et al. (2014).

108

6.2

B. Shen et al.

Rock Fracture Propagation Mechanisms and Fracture Criterion

Fractures in rock masses often exhibit different behaviours from those in other engineering materials such as steel and glass. Fractures in underground rock masses are mostly under compression, and they may propagate in tensile, shear or tearing modes. Unlike in other materials, shear fracture is one of the common types of fractures observed in the rock mass, caused by high compressive stresses existing in deep underground. Although the tensile strength of rock is much less than its shear strength, under compressive and confined conditions in underground rock masses, tensile stresses may not always exist and hence rock mass failure is often dominated by shear fracture initiation and propagation. A number of laboratory tests have been conducted to investigate the fracture propagation mechanisms in compression using brittle geo-materials. Petit and Barquins (1988) used a sandstone specimen with a single pre-existing fracture which was loaded in uniaxial compression. They observed two types of fractures: wing cracks approximately in the direction of loading and secondary cracks in the direction of about 45
from the loading direction. Shen et al. (1995) used specimens made of artificial rock with two parallel and aligned pre-existing fractures and applied a uniaxial load until failure. It was found that the rock bridge between the two pre-existing fractures failed due to shear fracture propagation which was initiated at the tips of the pre-existing fractures. The shear fracture is clearly recognisable by its rough surfaces with associated pulverised materials from the shearing of asperities, compared with the tensile crack (wing crack) which had clean and smooth surfaces. Rao (1999) carried out several shear tests on rock specimens with a single pre-existing fracture and found that, with a sufficiently high normal stress, the pre-existing fracture propagated in its original plane dominated by a shear failure mode. Backers et al. (2002) developed a method of testing the shear fracture toughness by using a punch-through test set-up of a pre-slotted core specimen. With high confinement stress, they found that shear fractures will develop and propagate in the rock bridge between the aligned slots. All these laboratory and field observations demonstrate that shear fracturing does occur in rocks with confinement when loaded in compression. In modelling fracture propagation in rock masses where both tensile and shear failure are common, a fracture criterion for predicting both tensile (mode I) and shear (mode II) fracture propagation is needed. The exiting fracture criteria in the macroapproach can be classified into two groups: the principal stress (strain)-based criteria and the energy-based criteria. The first group consists of the Maximum Principal Stress Criterion and the Maximum Principal Strain Criterion; the second group includes the Maximum Strain Energy Release Rate Criterion (G-criterion) and the Minimum Strain Energy Density Criterion (S-criterion). The principal stress (strain)-

6 Modelling Rock Fracturing Processes with FRACOD

G

Original surface

New surface

(a)

=

GI

109

+

GII

Growth

(b)

(c)

Fig. 6.1 Definition of GI and GII for fracture growth. (a) G, the growth has both open and shear displacement; (b) GI, the growth has only open displacement; (c) GII, the growth has only shear displacement

based criteria are only applicable to the mode I fracture propagation which relies on the principal tensile stress (strain). To be applied for the mode II propagation, a fracture criterion has to consider not only the principal stress (strain) but also the shear stress (strain). From this point of view, the energy based criteria seem to be applicable for both mode I and II propagation because the strain energy in the vicinity of a fracture tip is related to all the components of stress and strain. Both the G-criterion and the S-criterion have been examined for application to the mode I and mode II propagation (Shen and Stephansson 1993), and neither of them is directly suitable. In a study by Shen and Stephansson (1994) the original G-criterion has been improved and extended. The original G-criterion states that when the strain energy release rate in the direction of the maximum G-value reaches the critical value Gc, the fracture tip will propagate in that direction. It does not distinguish between mode I and mode II fracture critical energy (GIc and GIIc). In fact, for the most of the engineering materials, the mode II fracture toughness is much higher than the mode I toughness due to the differences in the failure mechanism. Laboratory-scale tests of rocks, for instance, found GIIc to be at least two orders of magnitude higher than GIc (Li 1991). It is difficult to apply the G-criterion to the mixed mode I and mode II fracture propagation since the critical value Gc must be carefully chosen to be between GIc and GIIc. A modified G-criterion, namely the F-criterion, was proposed (Shen and Stephansson 1994). Using the F-criterion the resultant strain energy release rate (G) at a fracture tip is divided into two parts, one due to mode I deformation (GI) and the other due to mode II deformation (GII). Then the sum of their normalized values is used to determine the failure load and its direction. GI and GII can be expressed as follows (Fig. 6.1): if a fracture grows a unit length in an arbitrary direction and the new fracture opens without any surface shear dislocation, the strain energy loss in the surrounding body due to the fracture growth is GI. Similarly, if the new fracture

110

B. Shen et al.

has only a surface shear dislocation, the strain energy loss is GII. The principles of the F-criterion can be stated as follows: 1. In an arbitrary direction (θ) at a fracture tip there exists a F-value, which is calculated by F ðθÞ ¼

GI ðθÞ GII ðθÞ þ GIc GIIc

ð6:1Þ

2. The possible direction of propagation of the fracture tip is the direction (θ ¼ θ0) for which the F-value reaches its maximum. F ðθÞjθ¼θ0 ¼ max :

ð6:2Þ

3. When the maximum F-value reaches 1.0, the fracture tip will propagate, i.e. F ðθÞjθ¼θ0 ¼ 1:0

ð6:3Þ

The F-criterion is actually a more general form of the G-criterion and it allows us to consider mode I and mode II propagation simultaneously. In most cases, the F-value reaches its peak either in the direction of maximum tension (GI ¼ maximum while GII ¼ 0) or in the direction of maximum shearing (GII ¼ maximum while GI ¼ 0). This means that a fracture propagation of a finite length (the length of an element, for instance) is either pure mode I or pure mode II. However, the fracture growth may oscillate between mode I and mode II during an ongoing process of propagation, and hence form a path which exhibits the mixed mode failure in general.

6.3

Theoretical Background of FRACOD

The FRACOD code is essentially a Boundary Element Method (BEM) program and thus it follows the BEM principals. Specifically, it utilizes the Displacement Discontinuity Method (DDM) which is an indirect boundary element technique. The fracture mechanics theories and the F-criterion are incorporated into the code to model fracturing process. The DDM employed in FRACOD is based on the analytical solution of stresses and displacements caused by a constant displacement discontinuity over a finite line

6 Modelling Rock Fracturing Processes with FRACOD (b)

s

(a)

y

111

n

s n

α

n=0+

N

(x,y) j

j

2a

N

n=0-

Ds

j

j

Dn

x i

2a i

i

i

3 1 2

3 1 2

σs

σn

Fig. 6.2 Representation of a curved crack by N displacement discontinuity elements

segment (e.g. crack) in an infinite elastic solid body in 2D. The displacements in the solid body are continuous everywhere except over the line segment where they differ by a constant value which is defined as the displacement discontinuity. The explicit solution of the given problem was provided by Crouch and Starfield (1983). The stresses and displacements of a specific point can be found using Eq. (6.4). σ s ¼ Ass Ds þ Asn Dn σ n ¼ Ans Ds þ Ann Dn us ¼ Bss Ds þ Bsn Dn

ð6:4Þ

un ¼ Bns Ds þ Bnn Dn where Ds and Dn are the shear and normal components of displacement discontinuity. Ass, etc., and Bss, etc., are the boundary influence coefficients for the stress and displacement respectively. The coefficients are the functions of elastic properties of solid body and the position of the point relative to the line segment. They represent the stresses or displacements of the point caused by a constant unit displacement discontinuity. For a crack of any shape, it is acceptable to represent it by N straight segments, joined end by end as shown in Fig. 6.2, provided that the number of line segment is sufficient. For each line segment, an elemental displacement discontinuity exists (Djs and Djn). Based on the principal of superposition and applying Eq. (6.4), the stresses and displacements at any point in the infinite body can be obtained. Applying the

112

B. Shen et al.

expressions to points on the line segments along the crack, the stresses and displacements have the form σ is ¼ σ in ¼

N X

Aijss Djs þ

¼

uin ¼

Aijsn Djn

j¼1

j¼1

N X

N X

Aijns Djs þ

j¼1

uis

N X

N X

Bijss Djs

þ

for i ¼ 1 to N

N X j¼1

N X

N X

Bijns Djs þ

ð6:5Þ

Bijsn Djn

j¼1

j¼1

Aijnn Djn

j¼1

Bijnn Djn

j¼1

The equations compose a system of simultaneous linear equations with 2N unknowns. They are the elemental displacement discontinuity components Djn and Djs , and can be solved by applying appropriate 2N equations from Eq. (6.5) to the specified traction and/or displacement conditions on the crack. For a problem of simulating discontinuities in the rock mass, various constraints to the stress in the equations can be added: For an open crack where no stresses can be transmitted through it, the stress components in Eq. (6.5) are zero. σ is ¼ 0 σ in ¼ 0

ð6:6Þ

For the crack surface of elastic contact, the stress components depend on the crack stiffness (Ks, Kn) and the displacement discontinuities and have form σ is ¼ K s Dis σ in ¼ K n Din

ð6:7Þ

Coulomb’s failure is adopted for a crack with a sliding surface. σ is ¼ c þ K n Din tan ϕ σ in ¼ K n Din

ð6:8Þ

where ϕ is the crack friction angle and c is the cohesion strength. Finally, with the appropriate boundary stresses and displacements, the unknown elemental displacement discontinuities (Djn and Djs ) are obtained by solving the

6 Modelling Rock Fracturing Processes with FRACOD

113

system of stress governing equations in Eq. (6.5) using conventional numerical techniques for linear equations. The key step in using the F-criterion is to determine the strain energy release rate of mode I (GI) and mode II (GII) at a given fracture tip. As GI and GII are only the special cases of G, the problem is then how to use DDM to calculate the strain energy release rate G. The G-value, by definition, is the change of the strain energy in a linear elastic body when the crack has grown one unit of length. Therefore, to obtain the G-value the strain energy must first be estimated. By definition, the strain energy, W, in a linearly elastic body is ZZZ W¼ V

1 σ ε dV 2 ij ij

ð6:9Þ

where σ ij and εij are the stress and strain tensors, and V is the volume of the body. The strain energy can also be calculated from the stresses and displacements along its boundary W¼

1 2

Z ðσ s us þ σ n un Þds

ð6:10Þ

s

where σs, σn, us, un are the stresses and displacements in tangential and normal direction along the boundary of the elastic body. Applying Eq. (6.10) to a single straight crack in an infinite body with far-field stresses in the shear and normal direction of the crack, (σs)0 and (σn)0, the strain energy, W, in the infinite elastic body is W¼

1 2

Z 0

a 

    σ s  ðσ s Þ0 Ds þ σ n  ðσ n Þ0 Dn Þda

ð6:11Þ

where a is the crack length, Ds is the shear displacement discontinuity and Dn is the normal displacement discontinuity of the crack. When DDM is used to calculate the stresses and displacement discontinuities of the crack, the strain energy can also be written in terms of the element length (ai) and the stresses and displacement discontinuities of the ith element of the crack. Then the strain energy from the whole crack is the sum of the energies of all elements      i  i i i i i 1X i i a σ σs W Ds þ a σ  σ Dn Þ 2 i s n n 0 0

ð6:12Þ

It is noted that the far-field stresses are resolved along the elemental directions for each element.

114

B. Shen et al.

Fig. 6.3 Fictitious crack increment Δa in direction θ with respect to the initial crack orientation

Crack

Δa

θ

a The G-value in the direction θ at a crack tip can be estimated by G ðθ Þ ¼

∂W ½W ða þ ΔaÞ  W ðaÞ  Δa ∂a

ð6:13Þ

where W(a) is the strain energy governed by the original crack, and W(a+Δa) is the strain energy governed by a new crack consisting of the original crack and a small extension of Δa at the crack tip in the direction of θ (Fig. 6.2). In Fig. 6.3, a ‘fictitious’ element is introduced to the tip of the original crack with the length Δa in the direction θ. Both W(a) and W(a+Δa) can be determined easily by directly using DDM and Eq. (6.12). In the above calculation, if we restrict numerically the shear displacement of the “fictitious” element to zero, the result obtained using Eq. (6.13) will be GI(θ). Similarly, if we restrict the normal displacement of the “fictitious” element to zero, the result obtained will be GII(θ). After obtaining both GI(θ) and GII(θ), the F-value in Eq. (6.1) can be calculated using the given fracture toughness values GIc and GIIc of a given rock type. Then the maximum value of F and the corresponding direction can be sought. If the maximum value is greater than unity, then the crack will propagate in the maximum direction at the tip.

6.4

Coupling Between Rock Fracturing and Thermal and Hydraulic Processes

Over the past several decades, coupled Mechanical–Thermal–Hydraulic processes in rock masses have been a focus of research, particularly in the field of underground nuclear waste disposal, and significant advances have been achieved (Min et al. 2005; Rutqvist et al. 2005; Tsang et al. 2005). However, the past studies have mostly treated the rock mass as a continuum or a discontinuum with predefined discontinuities. The process of explicit rock fracturing, which is the dominant mechanism in hard rock failure, has not been adequately addressed during the simulation of complex coupled processes. Understanding and predicting the effects of the interactive processes between explicit rock fracturing, temperature change and fluid flow (coupled Fracturing (F) – Thermal (T) – Hydraulic (H) processes) remain to be a key challenge for industries such as geothermal energy extraction, geological CO2

6 Modelling Rock Fracturing Processes with FRACOD

115

su es

es

ca us

es

Fl

ui

d

pr

tu r ac Fr

T t h e mp er e m ra a l tu s t re r e ch ss a es ng e

re

en

dr

ha

iv

nc

e

es

flu

fr a

id

ct ur

flo

es

w

Rock fracturing

Fluid flow change rock temperature

Fluid flow Temperature changes fluid viscosity

Temperature change

Fig. 6.4 Interaction between rock fracturing and fluid flow and rock temperature changes

sequestration, underground LNG storage, and deep geological disposal of nuclear waste. In a fractured rock mass, rock fracturing, fluid flow and rock temperature change are closely correlated (Fig. 6.4). Rock fractures will enhance the fluid flow by creating new flow channels and/or widening the channels, whereas the fluid pressure may stimulate fracture growth. A temperature change will result in thermal stress in the rock mass which could lead to fracture propagation. Secondary interaction among the three processes also exists: for example, fluid flow temperature, which in turn affects the rock stress state and may cause fracture propagation. Coupling between these processes in numerical modelling are necessary for the industrial applications mentioned previously. To increase the knowledge on the above mentioned issues and to understand the coupled F–T–H processes in rocks on an engineering scale, recent development of FRACOD has been focused on the coupled processes between rock fracturing and thermal and hydraulic processes. This section summarises theoretical background and numerical considerations of the coupled F–T–H functions in FRACOD.

6.4.1

Rock Fracturing – Thermal Coupling

The direct coupling between rock fracturing and thermal processes is a one-way coupling with stresses dependent on the temperature field following thermo-elasticity principles. As the DD method used in FRACOD is an indirect boundary element method, an indirect method is also used to simulate the temperature distribution and thermal stresses due to internal and boundary heat sources. With this method, fictitious heat sources with unknown strength over the boundary of domain are used, and it is therefore easier to consider the problem with internal heat sources (Shen et al. 2013). The two-dimensional fundamental solutions for temperature,

116

B. Shen et al.

stresses and displacements induced by a point heat source with unit strength located at the origin of the coordinate system in thermo-elasticity are given below.   1 Ei ξ2 4πk 

  2  2 Eα 2x2 1  eξ 1 2 σ xx ¼  Ei ξ 8πkð1  νÞ r ξ2

 2 Eα 2xy 1  eξ  2 σ xy ¼ 8πk ð1  νÞ r ξ2 

  2  2 Eα 2y2 1  eξ  Ei ξ 1 2 σ yy ¼ 8πkð1  νÞ r ξ2 8  9 2 < 1  eξ = α ð1 þ ν Þ x 1  2 þ Ei ξ ux ¼ r 2 4πk ð1  νÞ :r ; 2ξ2 T¼

8  9 2 < 1  eξ = α ð1 þ ν Þ y 1  2 uy ¼ Ei ξ þ r 2 4πk ð1  νÞ :r ; 2ξ2

ð6:14Þ ð6:15Þ ð6:16Þ ð6:17Þ

ð6:18Þ

ð6:19Þ

where T is the temperature (
C), σ xx, σ xy, and σ yy are the stresses (Pa), ux and uy are the displacements (m), α is the linear thermal expansion coefficient (1/
C), k is the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
thermal conductivity (W/m C), ν is the Poisson’s ratio, r ¼ x2 þ y2 , ξ2 ¼ 4cr r t , cr ¼ ρ kcp, cr is the thermal diffusivity (m2/s), with ρr being the density (kg/m3) and cp r R 1 z being the specific heat capacity (J/kg
C), t is time (s), and EiðuÞ ¼ u ez dz in the above equations. Equations (6.14, 6.15, 6.16, 6.17, 6.18, and 6.19) constitute the fundamental equations to be used in all the formulations of the numerical process for F-T coupling problems. For an internal problem as shown in Fig. 6.5, the boundary of a finite body is discretized into n elements. Before any mechanical boundary condition is considered, each element is assumed to be in an infinite, isotropic and homogeneous medium to make use of the above fundamental solutions. Let’s consider that a constant line heat source with unit heat strength is placed along element j at time t0 ¼ 0. At any given time t, the temperature, stresses and displacements at the centre point of another element (element i) is known based on the fundamental solutions given in Eqs. (6.14, 6.15, 6.16, 6.17, 6.18, and 6.19). In the fictitious heat source method, it is assumed that a line heat source has been applied along each boundary element. The strengths of these line sources are the unknowns and need to be solved for. The total temperature change, stresses and displacements at element i due to the fictitious line sources and mechanical boundary

6 Modelling Rock Fracturing Processes with FRACOD

117

Fig. 6.5 Elements along a solid body boundary and local coordinate system of the boundary element

conditions can be calculated by superimposing the effect of all individual heat sources as shown below: Ti ¼

n X

ð6:20Þ

T ij H j

j¼1

σ is ¼

n  X

Aijss Djs þ Aijsn Djn þ F ijs H j



ð6:21Þ

j¼1

σ in ¼

n  X

Aijns Djs þ Aijnn Djn þ F ijn H j



ð6:22Þ

j¼1

uis ¼

n  X

Bijss Djs þ Bijsn Djn þ Gijs H j



ð6:23Þ

j¼1

uin ¼

n  X

Bijns Djs þ Bijnn Djn þ Gijn H j



ð6:24Þ

j¼1

where H j is the strength of the line heat source at element j. Tij, Assij, Asnij, Ansij, Annij, Bssij, Bsnij, Bnsij, Bnnij, F ijs , F ijn , Gijs , Gijn are ‘influence coefficients’, representing the temperature, stress and displacement at the centre of the element i due to a unit line source (thermal and mechanical) at element j. They are calculated based on the Eqs. (6.14, 6.15, 6.16, 6.17, 6.18, and 6.19) and those in Eq. (6.4). For example, the coefficient Ansij gives normal stress at the midpoint of the ith element (σ in ) due to a constant unit shear displacement discontinuity over the jth element (Djs ¼ 1Þ.

118

B. Shen et al.

Because the strength (H j) of the fictitious heat sources is only dependent upon the thermal conditions, they can be solved separately by only using thermal boundary conditions. If the temperature along the problem boundary is known, using Eq. (6.20), there will be n equations with n unknowns. The fictitious heat source strength along each element can then be obtained by solving the system of n linear equations. Their values can then be used in Eqs. (6.21, 6.22, 6.23, and 6.24) to solve for the displacement discontinuities Djs and Djn . Alternatively, heat flux rather than the temperature may be prescribed on part or whole of the boundary. In this case, the flux condition is used to replace Eq. (6.20) for temperature. The heat flux in the normal direction of element i due to a unit line source at element j is given by Eq. (6.25)     xi  xj cos θi þ yi  yj sin θi  2  ∂T ¼ Ei ξ Qij ¼ k ∂n 8πkt 2

ð6:25Þ

The basic principle of the indirect boundary element approach for thermo-elastic analysis is the assumption that a fictitious line heat source exists at each element. The strengths of the line sources are unknown and should be determined based on the boundary conditions. For example, if the temperature at all boundary elements is zero, the combined effect of all the line heat sources on the boundary elements should result in a zero temperature. Once the strength of each fictitious heat source is determined, the temperature, thermal flux, and thermal-induced stresses and displacements at any given location in the rock mass can be calculated. By applying mechanical boundary conditions, the displacement discontinuities on the elements can also be determined and the total stresses and displacements at any position can then be calculated.

6.4.2

Fracturing – Hydraulic Flow Coupling

In fractured hard rock such as granite, fluid flow occurs predominantly through explicit fractures rather than through intact rock due to the low permeability of the intact rock. Fluid pressure in rock fractures may cause rock fracture movement, increase fracture aperture or even cause fracture propagation. On the other hand, fracture movement and propagation will change the fracture hydraulic conductivity and create new flow paths. The two way interaction between fracture mechanical response and fluid flow is critically important in studying the coupled fracturing – hydraulic flow (F-H) processes. Two fundamental approaches have been used in modelling the hydro-mechanical coupling in fractured rock medium. The first is the implicit approach, where fluid flow equations are solved together with mechanical equations for rock matrix and fractures. Most of the finite element codes designed for modelling the porous flow using Darcy’s law are based on this approach.

6 Modelling Rock Fracturing Processes with FRACOD Fig. 6.6 Domain division for fluid flow simulation

119

Domain 1 Domain 2 Domain 3 Q12 Q1r

Element 1 Fig. 6.7 Iterative process for a coupled F-H process

P

Q23 Q3r

Element 2 Element 3 fluid flow between domains & leakage into rock

domain fluid pressure change

fracture deformation

domain fluid pressure change

domain geometry change

The second is the explicit approach, where both fluid flow and mechanical response are simulated using a time marching iteration process. The well known commercial code UDEC by Itasca (2004) is based on this approach. The explicit approach is mathematically simpler and easier to adopt the complicated (and evolving) model boundary conditions than the implicit approach. However it often requires significantly longer computational time as small time steps are required to achieve convergence for the flow solution. The explicit approach is used in FRACOD. The mechanical calculation (including rock deformation and fracture propagation) employs the DDM with an iteration scheme for modelling fracture propagation processes. The fracture fluid flow calculation is conducted through the time-marching iteration based on the cubic law (Louise 1969) derived predominantly for fluid flow in rock fractures. However, leakages from fracture channels to the rock matrix are also considered. During the mechanical numerical simulation using the DD method, a fracture is discretised into a number of DD elements. In the flow calculation, each DD element is considered as a hydraulic domain and adjacent domains are connected hydraulically (see Fig. 6.6). Fluid may flow from one domain to another depending on the pressure difference between the two domains. The solution of a coupled F-H problem can be achieved numerically using the iteration scheme shown in Fig. 6.7, and the iteration steps are described below. Step 1. Fluid flow occurs between fracture domains and fluid leaks into the rock matrix. The fluid flow between fracture domains is calculated using the Cubic Law. The flow rate (Q) between two domains is calculated using Eq. (6.26): Q¼

e3 ΔP 12μ l

ð6:26Þ

120

B. Shen et al.

where e is fracture hydraulic aperture of the element domain; l is element length; ΔP is fluid pressure difference between the two element domains; and μ is fluid viscosity. The leakage from a fracture domain into the rock matrix is calculated using Eq. (6.27). Qleak ¼

k w P  P0 d μ

ð6:27Þ

where kw is rock permeability; d is effective leakage distance; P is domain fluid pressure; P0 is initial pore pressure. The effective leakage distance d is based on the assumption that at distance of d from the fracture surface, the fluid pressure equals to the initial pore pressure. Obviously, the effective leakage distance is closely related to the flow time and the fracture system configuration. In the case of a long fracture with a constant fluid pressure inside the fracture, the effective leakage distance may be estimated using one-dimensional porous flow equations and it varies with flow time. For the case with irregular fracture systems, accurate estimate of the effective leakage distance will be much more difficult. In this case, one may consider this option as a rough guide only. Step 2. Fluid flow causes changes in domain fluid pressure. The new domain pressure due to fluid flow during a small time duration Δt is calculated using Eq. (6.28): Pðt þ Δt Þ ¼ P0 þ Ew Q

Δt Δt  Ew Qleak V V

ð6:28Þ

where Ew is the fluid bulk modulus; V is domain volume. Step 3. Change in fluid pressure causes fracture deformation. The fracture deformation is calculated using the DD method where the new fluid pressures in fracture domains are the input boundary stresses. After considering the fluid pressure in the fracture domains (elements), the system of equations for calculating the element displacement discontinuities is given in Eq. (6.29): 8  N ij i N ij i i P P i > > ¼ AD þ AD  K s D > < σs ss s sn n s 0

j¼1

j¼1

N ij i N ij i i >i P P > > AD þ AD  K nD : σ þ Pðt þ Δt Þ  P0 ¼ n 0

j¼1 ns s

j¼1 nn n

ð6:29Þ

n

During this step, the additional fracture deformation caused by any fracture propagation has also been considered and incorporated into the solutions. Step 4. Fracture deformation changes the domain volume, and hence changes the fluid pressure in domains. The new domain pressure is calculated using Eq. (6.30)

6 Modelling Rock Fracturing Processes with FRACOD

P0 ðt þ Δt Þ ¼ Pðt þ Δt Þ  Ew

121

Δe  l V

ð6:30Þ

Here Δe is the change of the fracture aperture at the element. The new domain fluid pressures are then used to calculate the flow rate between domains in Step 1. Steps 1–4 are iterated until the desired fluid time is reached and a stable solution is achieved. During the fluid flow calculation, a proper time step is needed for the iteration process to converge to the final solution. The time step should meet the following condition Δt
> > > <   = G σ xy ðQÞ ¼ þ ð1  νÞI xz  zI xyy Dy 4π ð1  νÞ > > >  > ; :   ð1  2νÞI xy þ zI xyz Dz ( ) ½ð1  νÞI zz  νI xx  zI xxz Dx G σ xz ðQÞ ¼   4π ð1  νÞ  νI xy þ zI xyz Dy  zI xzz Dz (   )  νI xy þ zI xyz Dx G σ yz ðQÞ ¼   4π ð1  νÞ þ ð1  νÞI zz  νI yy  zI yyz Dy  zI yzz Dz

ð7:4Þ

Here G and v are the shear modulus and the Poisson’s ratio of the elastic material, respectively. The subscripts of I denote the partial derivatives, which are functions of the position (x,y,z). It is noted that the classical sign convention for normal stresses are employed here, that is, tensile normal stresses are positive. Using notations Bik(Q) ¼ Bik(x,y,z) (i,k ¼ x,y,z) for the coefficients of the displacement discontinuity components Dx, Dy, Dz in the displacement expressions and Aijk(Q) ¼ Aijk(x,y,z) (i,j,k ¼ x,y,z) for the coefficients in the stress components, the above solutions for point at Q(x,y,z) can be expressed in ui ðQÞ ¼

σ ij ðQÞ ¼

X

X

B ðQÞDk k¼x,y,z ik

i ¼ x, y, z

ð7:5Þ

A ðQÞDk k¼x,y,z ijk

i, j ¼ x, y, z

ð7:6Þ

The coefficients Bik(Q)and Aijk(Q) represent the influences to the ith displacement component and the ijth stress components of the kth unity DD component over the region. If the crack has a curvature or is large with non-uniform displacement discontinuity such that the conditions for the above expressions are invalid, then an approximation can be made. The approximation is made by subdividing the crack into small elements with simple planar shapes, such as triangles or rectangles, approximating the displacement discontinuity components on each element as uniform so that the above formula can be used, and then superposing the effects from all elements to

7 FRACOD3D: A Three-Dimensional Crack Growth Simulator Code

141

calculate the approximate value for the whole crack. Let Ne is the total number of the elements in the discretisation and Dek is the k (¼x,y,z) component of the displacement discontinuity on element e. It should be noted that the above solutions are expressed in the local coordinate system on each element. Before the superposition, the solutions from each element must be transformed so that they refer to a universal global coordinate system. Without confusion in the following, we will still use the coordinate system oxyz for the universal global coordinate system and let Beik ðQÞ and Aeijk ðQÞ (i,j,k ¼ x,y,z) be the coefficients of the DD components Dek on the element e in the displacement and stress component expressions referred to the universal coordinate system. Then the displacement and stress components at point Q(x,y,z), produced by the DD components on all the elements, are given by ui ðQÞ ¼ σ ij ðQÞ ¼

XN e X e¼1

XN e X e¼1

Be ðQÞDek k¼x,y,z ik

i ¼ x, y, z

ð7:7Þ

Ae ðQÞDek k¼x,y,z ijk

i, j ¼ x, y, z

ð7:8Þ

The above shows how to compute the displacement and stresses inside an infinite body that are caused by the displacement discontinuity on complicated cracks. In underground environment where in situ stresses exist, the right hand sides of (7.7) and (7.8) represent the displacement and stress, respectively, due to action on the cracks, so they are induced displacement and stress. The total displacement and stress are sum of them and the in situ values. Taking the in situ displacement as zero value, the in situ stress components as σ 0ij ðQÞ and still employing ui and σ ij to represent total displacement and stress components, respectively, then the total displacement components are still given by (7.7) and the total stress components are σ ij ðQÞ ¼

7.2.2

XN e X e¼1

Ae ðQÞDek k¼x,y,z ijk

þ σ 0ij ðQÞ i, j ¼ x, y, z

ð7:9Þ

Three-Dimensional Displacement Discontinuity Method

The above expressions are for displacement and stress components when the displacement discontinuity components on a crack in an infinite elastic body are known. However, in most practical problems related to cracks, the known variables in advance are displacements or tractions on the crack, not the DD components. In some other practical problems, the elastic body is finite, not infinite, in size, or there are cavities with finite volume in infinite body and the displacement and/or traction components on the boundary are generally given. All these problems can be solved using the DDM as outlined as follows.

142

J. Shi and B. Shen

With the DDM, problems are approximately solved with the following steps: 1. Discretise the crack into small planar elements. In the case of a finite body or an infinite body with a cavity, an infinite body with an imaginary crack is assumed. The shape of the imaginary crack is identical to the boundary of the finite body or the surface of the cavity in the infinite body. The imaginary crack is discretised into small planar elements. Assume that the number of the elements is Ne. 2. Approximate the specified continuously distributed displacements and/or tractions on the whole crack with a piece-wise continuous distribution, such as the uniform distribution, over each element. 3. Assume that the specified tractions and/or displacements on the crack are caused by some DD components over each planar element, and then try to determine the correct values of the DD components that give the known values of tractions and/or displacements. This is done by (a) equating the specified value of tractions and/or displacements at each element to the formulated expression in terms of the assumed DD components on all elements to set up a system of governing equations for the DD components and (b) solving the system of equations for the DD components. In the process of setting up the equations, variables must be transformed into a universal global coordinate system in which the tractions and/or displacements are given before the contributions from all the elements can be added together. 4. Calculate the displacements and the stresses inside the body with the determined values of the DD components according to the formulae given above. When the displacement components on the crack, or on the surface of exterior boundary of a finite body, or on surface of the cavity in an infinite body, are given as ubi ðx, y, zÞ, the governing equations for the DD components are obtained by equating the specified value to the right hand side of expression (7.8) for the collocating points Qm(xm,ym,zm) (m ¼ 1,2,. . .,Ne) at the centroid of each element where the displacements are specified, XN e X e¼1

Be ðQm ÞDek k¼x,y,z ik

i ¼ x, y, z,

¼ ubi ðQm Þ

m ¼ 1, 2, . . . , N e :

ð7:10Þ

Note that although the specified displacement components can refer to a global universal coordinate system, the displacement discontinuity components DDs refer to the local coordinate systems on the elements. For traction boundary value problems, we need to express tractions in terms of the displacement discontinuity components DD. With the stress components given in (7.9), we can obtain the traction components tx, ty, tz on a plane element at the point Q as t i ðQÞ ¼

XN e X e¼1

t e ðQÞDek k¼x,y,z ik

þ t 0i ðQÞ i ¼ x, y, z

ð7:11Þ

7 FRACOD3D: A Three-Dimensional Crack Growth Simulator Code

143

Here the coefficients t eik ðQÞ ði, k ¼ x, y, zÞ are related to the coefficients Aeijk ðQÞ and t 0i ðQÞ is the traction component due to the in situ stress. Equating the specified value to these expressions for collocating points at the centroid of each element where tractions are given, we can obtain the governing equations for the DD components, XN e X e¼1

t e ðQm ÞDek k¼x,y,z ik

¼ t bi ðQm Þ  t 0i ðQm Þ

i ¼ x, y, z, m ¼ 1, 2, . . . , N e

ð7:12Þ

If displacement is given on part of the boundary surface and traction is specified on the remaining part, then the problem has a type of mixed boundary conditions. In this case, the discretisation should make the boundary between the traction part and displacement part lie on the sides of the small elements. Employing the Eq. (7.10) for displacement on the specified displacement part and (7.12) for traction on the specified traction part will produce a system of equations for the DD components. There is another type of mixed boundary conditions: on an element, conditions are not specified for all three displacement components or not specified for all three traction components. It is that one or two displacement component/s and two or one traction components are given on some or all elements. Sliding of a body on another hard body is one example of this. In this case, expressions for the corresponding component in (7.10) and (7.12) should be used to establish the governing equations. With the DDM, linear equations for the DDs are first set up according to the given conditions on the collocating boundary elements. After DDs are solved from the equations, the displacements and stresses at other required points can be calculated. However, if the points are too close to or on the boundary elements, then the stresses calculated in this way have large errors. A numerical scheme has been proposed by Shi and Shen (2014) to calculate stresses on the elements. The material for which the classical DDM can be applied is linear elastic isotropic and homogenous. However the engineering materials are most likely not isotropic and homogeneous. In many engineering problems, the material is piecewise isotropic and homogenous in that the problem domain has several sub regions and within each of the regions, the material is isotropic and homogenous. For example, in coal mining, the coal seams are embedded between mass rocks with different properties. Shi and Shen (2013) outlined a scheme for solving three-dimensional problems with multiple material regions.

7.2.3

Verification of the DDM Code

We verify the code FRACOD3D with analytical solutions of two examples. As the first example, we investigate the problem with a planar circular crack in the xy-plane of an infinite elastic body under uniform tension in the z direction at far field. The Young’s modulus and Poisson ratio of the material is 5 GPa and 0.2, respectively.

144

J. Shi and B. Shen

Table 7.1 Normal displacement at elements near the x-axis r/a uz/a(103) (F) uz/a(103) (A) Difference %

0.131 2.416 2.424 0.33

0.327 2.320 2.310 0.42

0.524 2.114 2.083 1.49

0.720 1.753 1.696 3.36

0.917 1.102 0.976 12.89

The tensile stress at far field is 10 MPa perpendicular to the crack plane. Radius of the crack is 3 m. The crack was divided into 144 triangular elements with 81 nodes. The comparison of displacements on the crack surface normal to the crack plane calculated with FRACOD3D (F) and analytical solution (A) is shown in Table 7.1. The results of the analytical solution was computed from that given in Sneddon and Lowengrub (1969). The FRACOD3D underestimates the displacement near the centre of the crack, with very limited error, and overestimates the displacement away from the centre. The difference between results of the FRACOD3D and those of the analytical solution increases with the distance from the centre of the crack. Overestimation of displacement on the whole crack by DDM is observed by Kuriyama and Mizuta (1993) and Ishijima et al. (1980). Ishijima et al. (1980) observed that the displacement difference increases with the distance from the centre of the crack, while graphs in Kuriyama and Mizuta (1993) show the opposite. The second example considers deformation and stress state due to inner uniform pressure on the spherical cavity surface in an infinite body, for which the analytic radial displacement and normal stresses are given in Saada (1993). The material parameters used are E ¼ 1 GPa and ν ¼ 0.25. The radius of cavity is a ¼ 3 m and internal pressure is Pi ¼ 1 MPa. The mesh for FRACOD3D has 224 triangular elements and 114 nodes. The variations of radial normal stress and radial displacement along and close to the x-axis in the body from FRACOD3D (FRACOD) and the analytical solution are plotted in Fig. 7.2. The x-axial direction is along the radial direction, so we have ux ¼ uR andσ xx ¼ σ RR. But it is noted that the values at x/a ¼ 1 from FRACOD3D are calculated at the centre of an element, which is close to, but not on, the x-axis. It can be seen that the stress and displacement are very close to the analytical values.

7.3

Three-Dimensional Crack Growth Simulation

Cracks exist in natural rock and mineral media, so the growth of cracks should be considered for engineering problems related to the strength of rocks and minerals. Simulation of crack growth involves: the stress and strain analysis, crack growth detection and implementation of the growth evolution. There are many methods and software packages for stress and strain analysis. Crack growth detection considers crack propagation criteria, which determine three components: (1) when a crack or part of it will start to grow, (2) in what orientation it will evolve, and (3) how far it can propagate. Implementation of the crack growth evolution deals with how to form

7 FRACOD3D: A Three-Dimensional Crack Growth Simulator Code

145

radial normal stresses(MPa)

1.2

1 FRACOD Analytic

0.8

0.6

0.4

0.2

0 0

0.5

1

1.5

2

2.5

3

2.5

3

x/a (a)

radial displacement (x10-3m)

2.5

2

FRACOD Analytic

1.5

1

0.5

0 0

0.5

1

1.5

2 x/a

(b)

Fig. 7.2 Variation of analytical and DDM (a) radial normal stress and (b) radial displacement along the x-axis around a spherical cavity

146

J. Shi and B. Shen

the new crack front during the process of crack propagation. Three-dimensional crack growth has a very complicated mechanism and various criteria exist for crack growth detection, such as maximum principal stress criterion, minimum strain energy density factor criterion and maximum energy release rate criterion, see Kassir and Sih (1974) and Wilde (2000). One theory could determine one, or two, or all the three components of fracture propagation detection defined above.

7.3.1

Crack Growth Criteria with FRACOD3D

It is well known that, in three-dimensional problems, there are three basic modes at a crack front in which the crack could grow: open Mode I, shearing Mode II and tearing Mode III, depending on the stress state around the crack front. Tensile stress in a direction normal to the crack plane causes a crack growing under the open mode. The shearing and tearing modes are due to the shear stresses in different directions in plane of the crack. Each fracture mode has a corresponding basic stress intensity factor, which measures the intensity of stress singularity around the fracture front and can be used to determine whether or not a fracture will grow. The stress intensity factors can be calculated with the differences of the displacements on the two surfaces of fracture near the front, which are just the basic variables DDs computed from DDM outlined above. This is one of advantages of DDM in crack growth analysis. We employ a quasi two-dimensional criterion for the crack growth in the code FRACOD3D and assume that at a front element, the crack grows in the plane that contains the front edge of the growing element and along a direction in the plane normal to the front edge. This direction is determined as follows. On this normal plane, the three basic stress intensity factors can be calculated in terms of the DDs (Dn, Db, Dt) as rffiffiffiffiffi 2π ðDb Þ d rffiffiffiffiffi E 2π K II ¼ D 8ð1  ν2 Þ d n rffiffiffiffiffi E 2π D K III ¼ 8ð 1 þ ν Þ d t E KI ¼ 8ð1  ν2 Þ

ð7:13Þ

(see Wilde (2000) and Shi et al. (2014)). As defined in DDM, Db, the displacement discontinuity normal to the plane of the front element, is negative for displacements related to open crack surfaces. Dn and Dt are the displacement discontinuity components in the plane of the front element, which are perpendicular and tangential to

7 FRACOD3D: A Three-Dimensional Crack Growth Simulator Code

147

the front edge, respectively. Here d is the distance from the centre of the crack front element to the crack front. The stress state (in spherical coordinate) near the crack front around this normal plane due to three-dimensional deformation has the following dominate terms (see Bower 2010, neglecting terms of O(r) and higher orders)    

1 θ 3θ θ 3θ σ r ¼ pffiffiffiffiffiffiffi K I 5 cos  cos þ K II 5 sin þ 3 sin 2 2 2 2 4 2πr    

1 θ 3θ θ 3θ σ θ ¼ pffiffiffiffiffiffiffi K I 3 cos þ cos  K II 3 sin þ 3 sin 2 2 2 2 4 2πr    

1 θ 3θ θ 3θ τrθ ¼ pffiffiffiffiffiffiffi K I sin þ sin þ K II cos þ 3 cos 2 2 2 2 4 2πr K III θ τrϕ ¼  pffiffiffiffiffiffiffi sin 2 2πr K III θ τθϕ ¼ pffiffiffiffiffiffiffi cos 2 2πr

ð7:14Þ

Here r and θ are polar coordinates in this normal plane (ϕ ¼ 0). Note that in the basic crack growing modes,pthe ffiffiffiffiffiffiffiffiffistress intensity factors are the proportional factors of maximum stresses to 1/ 2π r . We now define the quantity that is the proportional factor of the maximum circumferential normal from the pffiffiffiffiffiffiffiffistress ffi three-dimensional combined opening deformation in (7.14) to 1/ 2π r as combined mode I stress intensity factor KIe,    

1 θ 3θ θ 3θ K I 3 cos þ cos  K II 3 sin þ 3 sin K Ie ¼ max 2 2 2 2 πθπ 4    

1 θIe 3θIe θIe 3θIe ¼ K I 3 cos þ cos  K II 3 sin þ 3 sin 4 2 2 2 2

ð7:15Þ

The angle θIe defines the direction in which the maximum value KIe occurs. Similarly, we define the quantity proportional to the maximum shear stresses as combined mode II and mode III stress intensity factors,      1  θ 3θ θ 3θ  K IIe ¼ max K I sin þ sin þ K II cos þ 3 cos 2 2 2 2  πθπ 4      1  θIIe 3θIIe θIIe 3θIIe  ¼ K I sin þ sin þ K II cos þ 3 cos  4 2 2 2 2     θ  θ   K IIIe ¼ max K III cos  ¼ K III cos IIIe  ¼ K III 2 2 θ2½π, π 

ð7:16Þ

ð7:17Þ

with θIIIe ¼ 0. The toughnesses KIIc and KIIIc measure the material limits to crack growth under pure shearing deformation modes II and III. The internal friction and

148

J. Shi and B. Shen

cohesion add strengths to resist the crack growing when there is compressive normal stress near the crack front. So we can set up combined toughnesses for shear deformation modes as:

o pffiffiffiffiffiffiffin K IIce ¼ K IIc þ lim 2πr σ θ jθ¼θIIe tan φ þ c

ð7:18Þ

o pffiffiffiffiffiffiffin K IIIce ¼ K IIIc þ lim 2πr σ θ jθ¼θIIIe tan φ þ c

ð7:19Þ

r!0

r!0

with ϕ being the internal friction angle and c being the cohesion of the material. (Note that tensile normal stress is positive and compressive normal stress is negative. Tensile normal stress will reduce the pffiffiffiffiffiffiffiffi ffi thoughnesses.) Since the term c does not have a singularity, its product with 2π r is neglected and we have

K IIce

 9 8  θIIe 3θIIe > > > > > = < K I 3 cos 2 þ cos 2  > 1 tan φ ¼ K IIc    4> > > > θ 3θ > ; : K II 3 sin IIe þ 3 sin IIe > 2 2 K IIIce ¼ K IIIc  K I tan φ

ð7:20Þ

ð7:21Þ

since θIIIe ¼ 0. To determine the direction and mode for the crack propagation, we define the ratios: RI ¼ K Ie =K Ic ,

RII ¼ K IIe =K IIce ,

RIII ¼ K IIIe =K IIIce :

ð7:22Þ

Let RC be the maximum of RI, RII and RIII. If Rc < 1, then the crack does not propagate. If Rc  1 and Rc ¼ RI (RII or RIII), then it is regarded as mode I (II or III) propagation and the direction is given by the angle θIe(θIIe or θIIIe).

7.3.2

Implementation of Front Evolution of an Internal Crack

The matter of how far the crack would propagate is handled in incremental way as follows. For the steady-state problem, it is considered that the propagation distance is “one element length”. If some front element propagates, new elements with similar size are added to its front edge, depending on the size of the element. If there are many elements propagating, new elements are added to each of the propagating front elements. After the new elements and new nodes are added and proper boundary conditions are enforced for the new system, a new calculation for the displacement discontinuities and stresses is performed. The propagation criterion is checked again

7 FRACOD3D: A Three-Dimensional Crack Growth Simulator Code

149

with the new results. If the cracks with the new configuration continue to grow at a front element, then new element is added at the front element again. This process continues until growth stops on all the crack front elements. Now we consider the implementation process of the crack front evolution. After the displacement discontinuities on the cracks (as well as the displacement and stress at interior points) are obtained, all the crack front elements are checked for propagation, including the determination of propagation orientations, according to the criterion outlined in the previous section and this information is recorded. Then in the real implementation of the growth evolution, the front elements are traversed one-by-one and new elements are added to those that grow. Besides the normal meshing information between nodes and elements, FRACOD3D also needs information between elements and crack or surfaces, i.e., whether an element is on body surface or on a crack and on which crack if there are multiple cracks. With these information, FRACOD3D can find, and sort in order, the front elements of cracks. It can also determine whether a crack is isolated (internal crack) or joined to a body surface. In the case of internal crack, the front elements form a close ring. For the following explanation, the traverse is anticlockwise along the crack front and the terms “previous” and “next” neighbouring front elements will be used. The common front node with the previous neighbouring front element is termed as the first front node and the other one, common with next neighbouring front element, as second front node. It is noted that one unique propagation orientation is determined for a front element and the new elements can only be added at one front edge of the front element. So it is a requirement that the initial discretization must make front elements have only one front edge to avoid ambiguous scenario. This requirement should also be enforced when the new elements are created for joint growth. It is noted that triangular elements are employed in FRACOD3D. The displacement discontinuities are calculated for elements and the growth criteria are tested for the elements. So the implementation outlined here is element based, not node based, as in Krysl and Belytschko (1999) and Fries and Baydoun (2012). The number and layout of new elements for a growing front element depend on its size and propagation status of its neighbouring elements. There are many different situations to be considered. Generally for a growing front element, two new nodes are predicted and added for its two front nodes. If the next neighbouring front element grows, then the second predicted new node is replaced by the first new node of the next neighbouring front element to eliminate the gap, which exists if the two front elements are implemented independently. In this case, the implementation is similar to the node based scheme such as in Krysl and Belytschko (1999) and Fries and Baydoun (2012), and Fig. 7.3 illustrates an implementation of forming elements. The old front elements are those with solid edges and the new elements are those with dot, dash and dot-dash edges. The elements with dash edges can be regarded as results of growth of element C and those elements with dot-dash edges as growth of element D. The elements with dot edges are due to non-growing front element A and growing front element B, in which case, an extra element is added in front of non-growing front element A to smooth the sharp corner.

150

J. Shi and B. Shen

Fig. 7.3 Illustration of new elements (with dot, dash and dot-dash edges) from growing front elements B, C and D (with solid edges) and non-growing element A D

C B

A

Fig. 7.4 Small element B growing with non-growing neighbours A and C at both sides

C B

A

If the next neighbouring front element does not grow, then an extra element is added in front of it to smooth the sharp corner and eliminate the ambiguity of two front edges of one front element, as that shown in Fig. 7.4. Figure 7.4 illustrates the situation in which the front element B grows while its previous and next neighbouring front elements A and C do not grow. In this case, if we just add two new front nodes and link them to old front node of element B, then one of the new elements will have ambiguous two front edges and sharp corners are created with the non-growing neighbours. The ambiguity can be eliminated by adding new element in front of its neighbour A. However, it is not always suitable to add the extra element in front of non-growing front element. For example, if the two front edges of growing and non-growing front elements form an acute angle at the common front node, then adding an extra element will create an overlap region. In such case, the extra elements, especially the ambiguous front element, should not be added.

7 FRACOD3D: A Three-Dimensional Crack Growth Simulator Code

151

As the crack grows, its front will get longer and longer and the newly added elements will get larger and larger, if there is no mechanism to control the size. The elements should not be too big – front edge not too long – from the accuracy point of view, so we need a scheme to avoid too large elements being added. This is done by adding an extra node at the middle of edge between the two new nodes, if the front edge is too long. This will reduce the front element size. From simulations of some examples, it was found that for a growing front element, if only the propagation orientation determined for the front element is used, then the growth configuration could be very unsmooth: propagations of neighbouring elements could go into very different orientations with the potential for twisting or overlapping of the new elements. This has been noticed by Vandamme and Curran (1989). With some initial trials when the code FRACOD3D was first published (Shi et al. (2014)), it was found that these unsmooth effects could be reduced if the propagation orientation of a front element is modified by taking into account of propagation orientations of its neighbouring front elements. The modified propagation orientation of a front element combines the propagation orientations of three front elements: the considered front element and its immediate neighbours at both sides. In the combination, the importance of each element was controlled by weighting factors. The weighting factors were chosen as 0.6 and 0.2 for considered front element and its neighbours. Then we found that Ren and Guan (2017) smoothed the growth in two steps. First step is to smooth the basic directions of crack front at the considered front node, i.e. the directions t, n, b at the node used in Eq. (7.13). The directions t and b are taken to be the weighting average of the tangential directions of front edges and normal directions of element planes of the node’s two neighbouring front elements, respectively. The direction n follows the right hand rule. The second step is to smooth the stress intensity factors. The stress intensity factors at a front node is taken to be the weighting averages of the stress intensity factors of nodes around the considered node. In FRACOD3D, although the stress intensity factors and propagation orientations are determined for the front elements, the new nodes are to be added first based on the front nodes. Thus we include Ren and Guan’s (2017) two steps in the smoothing process. The overall smoothing procedure consists of four steps: 1. Similar to Ren and Guan’s (2017) procedure, the basic directions t and b are taken to be the weighting average of the tangential directions of front edges and normal directions of element planes of the node’s two neighbouring front elements, respectively. The direction n follows the right hand rule. 2. Instead of calculating the weighting averages of the basic stress intensity factors, we calculate weighting averages of the displacement discontinuities of the neighbouring elements and then use the averages to calculate the basic stress intensity factors. From the expressions in (7.13), this is very close to Ren and

152

J. Shi and B. Shen

Guan’s (2017) procedure – difference due to the factor d. The weighting average of displacement discontinuity D for a general front element I is calculated according to DI ¼ ð0:25DI2 þ 0:5DI1 þ DI þ 0:5DIþ1 þ 0:25DIþ2 Þ=2:5,

ð7:23Þ

where I  2, I  1, I + 1 and I + 2 represent the neighbours at two sides of the front element I and DI is the weighting average. With the weighting averages of the displacement discontinuities, stress intensity factors are evaluated and the growth orientations are determined for each front element. 3. As in Shi et al. (2014), the growth orientation θ of the front element I is smoothed by taking θI ¼ 0:2θI1 þ 0:6θI þ 0:2θIþ1 :

ð7:24Þ

In a similar manner, the incremental distance for the new elements can also be modified based on the neighbouring front elements. With the determined orientation, incremental distance and the basic directions of the crack surface at the front elements, the positions of new nodes can be predicted. 4. For further smoothing, the predicted positions of the new nodes around the considered front element/node are used to fit a quadratic or cubic threedimensional curve by the least square method and the smoothed node is a point on the fitting curve. In this process, the 3D curve is expressed in parameter form. For example, the x-coordinate of the points on a quadratic curve segment can be expressed as x ¼ xðt Þ ¼ a þ bt þ ct 2

ð7:25Þ

where 0  t  1 is the parameter and a, b and c are the coefficients, which is to be determined by least square fitting method. Let N new nodes distributed at two sides of the considered node I be fitted to the curve with parameter values 0¼t1