Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling [1st ed.] 9789811571961, 9789811571978

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Yongliang Wang

Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling

Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling

Yongliang Wang

Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling

123

Yongliang Wang China University of Mining and Technology (Beijing) Beijing, China

ISBN 978-981-15-7196-1 ISBN 978-981-15-7197-8 https://doi.org/10.1007/978-981-15-7197-8

(eBook)

Jointly published with Science Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Science Press. ISBN of the Co-Publisher’s edition: 978-7-03-063308-8 © Science Press 2021 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The damage and fracture in rock with multiple physical fields coupling are crucial mechanical issues in practical engineering, such as coal mining, oil and gas exploitation, and civil engineering. Grasping the influence mechanisms and controlling, the induced disasters demand high precision and reliable solutions in the whole process of evolution. The innovative adaptive numerical algorithm and simulation analysis have obvious advantages in accuracy and reliability. Since Ph.D. and postdoctorate stages, the author of this book has been engaged in research using the adaptive algorithm and simulation analysis method to determine the structural vibrations and stability and to study the damages and fractures in rocks. After receiving the Ph.D. from Tsinghua University, the author has constantly been involved in postdoctoral research work during two postdoctorate stints, both in computational rock mechanics, one at Tsinghua University and the other at the China University of Mining and Technology (Beijing). The author is currently an associate professor in the Department of Engineering Mechanics at the China University of Mining and Technology (Beijing). Postdoctoral work has been fruitful for the author. The main contents of this book were prepared entirely based on the postdoctoral work and related research papers published in peer-reviewed journals; some of the relevant results are, however, rearranged as systematic monographs. This book is organized as follows. Chapter 1 introduces the research background and significance. In Chap. 2, the finite element algorithm for continuum damage evolution of rocks, considering hydro-mechanical coupling, is introduced. In Chap. 3, finite element analysis of continuum damage evolution and wellbore stability of transversely isotropic rocks, considering hydro-mechanical coupling, are introduced. In Chap. 4, finite element analysis of continuum damage evolution and inclined wellbore stability of transversely isotropic rocks, considering hydromechanical-chemical coupling, are introduced. In Chap. 5, the adaptive finite element algorithm for damage detection in non-uniform Euler-Bernoulli beams with multiple cracks, using natural frequencies, is introduced. In Chap. 6, adaptive finite element-discrete element analysis of multistage hydrofracturing in naturally fractured reservoirs, considering hydro-mechanical coupling, is introduced. In Chap. 7, adaptive finite element-discrete element analysis of multistage supercritical CO2 v

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fracturing and microseismic modelling, considering thermal-hydro-mechanical coupling, is introduced. In Chap. 8, the adaptive finite element-discrete element-finite volume algorithm for 3D multiscale propagation of the hydraulic fracture network, considering hydro-mechanical coupling, is introduced. Finally, Chap. 9 summarizes the main conclusions and outlooks of this book. The author gratefully acknowledges the financial supports from the National Natural Science Foundation of China (Grant Nos. 41877275 and 51608301), the China Postdoctoral Science Foundation (Grant Nos. 2018T110158, 2016M601170, and 2015M571030), the Open Fund of Tianjin Key Laboratory of Soft Soil Characteristics and Engineering Environment (Grant No. 2017SCEEKL003), the Fundamental Research Funds for the Central Universities, Ministry of Education of China (Grant No. 2019QL02), the Teaching Reform and Research Projects of Undergraduate Education, CUMTB (Grant No. J190701), and the Yue Qi Young Scholar Project Foundation, CUMTB (Grant No. 2019QN14). The author gratefully acknowledges the guidance and advice from the respectable tutors during the master, Ph.D., and postdoctorate stages, Prof. Yuan Si and Prof. Zhuang Zhuo of Tsinghua University, and Prof. Wu Jianxun and Prof. Ju Yang of the China University of Mining and Technology (Beijing). During the postdoctorate stage, the author visited several research centres for computational mechanics in famous foreign universities as visiting scholar; the author gratefully acknowledges the advice and comments from the collaborators, Prof. Li Chenfeng, Prof. Feng Yuntian, and Prof. D. Roger J. Owen of the Zienkiewicz Centre for Computational Engineering at Swansea University in the UK, Prof. David Kennedy and Prof. Frederic W. Williams of the Applied and Computational Mechanics Group at Cardiff University in the UK, John Cain, Melanie Armstrong, and Fen Paw at Rockfield Software Ltd. in the UK, and Prof. Robert L. Taylor of the Department of Civil and Environmental Engineering at the University of California, Berkeley, in the USA. The author also gratefully acknowledges the editors, Ms. Wang Yun and Ms. Jin Rong, at the Science Press in China, for providing many suggestions and much assistance on formatting modifications and typesetting adjustments for improving this manuscript. The key contents of this book, such as multiphysical fields coupling, continuum damage, fracturing, and adaptive algorithm, are crucial in computational mechanics in rocks and are challenging the researchers’ best knowledge. Further work on these fields is needed for both theoretical and practical engineering advancements. Because this book is restricted by the limited knowledge of the author, a few errors are unavoidable. The author hopes all that experts, scholars, and other readers of this book will provide helpful suggestions for the book’s improvement. Beijing, China May 2019

Dr. Yongliang Wang Associate Professor

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Background and Significances . . . . . . . . . . . . . . . 1.2 Numerical Investigations for Damage and Fracture in Rock Considering Multiphysical Fields Coupling . . . . . . . . . . . . 1.3 Research Aims and Contents of the Book . . . . . . . . . . . . . 1.3.1 Research Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Research Contents . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Finite Element Algorithm for Continuum Damage Evolution of Rock Considering Hydro-Mechanical Coupling . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Assignment of Petrophysical Heterogeneity . . . . . . . . . . . . . . . 2.3 Governing Equations with Hydro-Mechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Continuum Damage Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Finite Element Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . 2.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Finite Element Model of Heterogeneous Rock . . . . . . . . 2.6.2 Rock Damage Analysis Under Different Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Effective Stress Analysis by Hydro-Mechanical Coupling Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Finite Element Analysis for Continuum Damage Evolution and Wellbore Stability of Transversely Isotropic Rock Considering Hydro-Mechanical Coupling . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Finite Element Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . 3.3 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . 3.3.3 Finite Element Model and Adaptive Mesh Refinement for Wellbore Stability Analysis . . . . . . . . . . . . . . . . . . . 3.4 Damage Tensor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Damage Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Stress and Permeability with Damage . . . . . . . . . . . . . . 3.5 Wellbore Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Pore Pressure and Stress Analysis of Rock Surrounding Wellbore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Instability Analysis of Wellbore Failure Region . . . . . . . 3.6.3 Collapse and Fracture Pressure Computation . . . . . . . . . 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Finite Element Analysis for Continuum Damage Evolution and Inclined Wellbore Stability of Transversely Isotropic Rock Considering Hydro-Mechanical-Chemical Coupling . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Transverse Isotropy and Hydration Characterization . . . . . . . . . 4.2.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Hydration Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Finite Element Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . 4.4 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . 4.4.2 Finite Element Model and Adaptive Mesh Refinement for Wellbore Stability Analysis . . . . . . . . . . . . . . . . . . . 4.5 Damage Tensor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Damage Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Stress and Permeability with Damage . . . . . . . . . . . . . . 4.6 Wellbore Stability Analysis Based on Weak Plane Strength Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Pore Pressure and Stress Analysis of Rock Surrounding Wellbore with Hydro-Mechanical-Chemical Coupling . . 4.7.2 Petrophysical Heterogeneity and Chemically Active Effect Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

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4.7.3 Time-Dependent Collapse and Fracture Pressure Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Coordinate Systems and Stresses Transformation . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Adaptive Finite Element Algorithm for Damage Detection of Non-Uniform Euler-Bernoulli Beams with Multiple Cracks Based on Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Adaptive Approach for Damage Detection of Cracked Beams . 5.2.1 Formulation and Analogy of Cracked Beams . . . . . . . 5.2.2 Stop Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Adaptive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Finite Element Solution . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Error Estimation and Mesh Refinement . . . . . . . . . . . . 5.4 Newton-Raphson Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Damage Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Free Vibration Problems . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Damage Detection Problems . . . . . . . . . . . . . . . . . . . . 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Adaptive Finite Element-Discrete Element Analysis for the Multistage Hydrofracturing in Naturally Fractured Reservoirs Considering Hydro-Mechanical Coupling . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Adaptive Finite Element-Discrete Element Method for Hydraulic Fracturing . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . 6.2.2 Numerical Discretization . . . . . . . . . . . . . . . . . . 6.2.3 Fracture Propagation and Local Remeshing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Numerical Models and Procedure . . . . . . . . . . . . . . . . . 6.3.1 Fracturing Models in Perforated Horizontal Wellbore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Discrete Fracture Network Models for Naturally Fractured Reservoir . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Global Procedure for Hydraulic Fracturing . . . . .

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6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Case Studies of Multistage Hydrofracturing of Unfractured and Naturally Fractured Models . . 6.4.2 Hydrofracturing Fracture Networks . . . . . . . . . 6.4.3 Flowback and Gas Production . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Adaptive Finite Element-Discrete Element Analysis for the Multistage Supercritical CO2 Fracturing and Microseismic Modelling Considering Thermal-Hydro-Mechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Adaptive Finite Element-Discrete Element Method for Fracturing and Microseismicity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Numerical Methods Considering Thermal-HydroMechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Local Remeshing Strategy for Fracture Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Microseismicity Analysis by the Evaluation of Moment Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Numerical Models and Procedure . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Numerical Models for Multistage Fracturing . . . . . . . . . 7.3.2 Global Procedure for Fracturing Considering Thermal-Hydro-Mechanical Coupling . . . . . . . . . . . . . . 7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Fracturing Fracture Networks . . . . . . . . . . . . . . . . . . . . 7.4.2 Fracturing Fluid and Gas Flow . . . . . . . . . . . . . . . . . . . 7.4.3 Microseismicity Analysis for Damaged and Contact Slip Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Adaptive Finite Element-Discrete Element-Finite Volume Algorithm for Three-Dimensional Multiscale Propagation of Hydraulic Fracture Network Considering Hydro-Mechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Governing Equations Considering Hydro-Mechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Solid Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Fluid Flow in Fractured Porous Media . . . . . . . . . . . . 8.2.3 Fracture Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.3 Combined Finite Element-Discrete Element-Finite Volume Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Computation Scheme for Hydro-Mechanical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Numerical Discretization for Solid Deformation . . . . 8.3.3 Cell-Centred Finite Volume Method for Fluid Flow . 8.4 Adaptive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Error Estimation for Stress Solutions . . . . . . . . . . . 8.4.2 Local Mesh Refinement . . . . . . . . . . . . . . . . . . . . . 8.4.3 Global Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Three-Dimensional Fracture Network Propagation of Engineering Scale Model . . . . . . . . . . . . . . . . . . 8.5.2 Three-Dimensional Fracture Network Propagation of Laboratory Scale Model . . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Chapter 1

Introduction

1.1 Research Background and Significances The multiple physical fields coupling and the damage and fracture of rocks are crucial mechanical issues in practical engineering, such as coal mining, oil and gas exploitation, and civil engineering (Krajcinovic and Lemaitre 1987; Abousleiman and Nguyen 2005; Peirce and Siebrits 2005; Nguyen and Abousleiman 2009; Watanabe et al. 2012; Bažant et al. 2014; Jin and Zoback 2017). However, understanding the influence mechanisms and controlling the induced disasters occurring due to damage and fracture evolution, require high-precision and reliable solutions. Continuous damage and fracture propagation are the main factors inducing burst, break and disaster in reservoir rocks (Marc-André et al. 2009; Erarslan and Williams 2012). However, few methods and technologies can capture the mechanisms of continuous damage and fracture propagation during the development of unconventional resources via well drilling and multistage fracturing. Furthermore, developing numerical simulation technology and understanding the mechanisms of unconventional drilling and fracturing can help provide better choices to predict, control and optimize the technologies for oil and gas extraction. Nevertheless, several crucial factors affect successful analysis to solve the above problems. Figure 1.1 shows the damage zone around a wellbore in a rock mass with bedding planes, where damage evolution accompanied drilling. The bedding planes influence damage evolution (Mortara 2010; Lee et al. 2012, 2013). The multiscale propagation of a fracture network of rock occurs in the case of horizontal well fracturing. The main and secondary fractures propagate on different scales, because the main fractures are affected by in situ stresses and are therefore bigger than the secondary fractures which are inter-connected via natural preexisting fractures. Figure 1.2 shows the multiphysical fields coupling of damage and fracture of rock. In certain types of some multiphysical field coupling (thermal, hydraulic, mechanical, and chemical), the thermal field induces temperature diffusion and affects the mechanical properties of the rock, the fluid pressure in porous media affects the solid deformation, the solid deformation affects the fluid flow, and © Science Press 2021 Y. Wang, Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling, https://doi.org/10.1007/978-981-15-7197-8_1

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1 Introduction

Fig. 1.1 Damage evolution around wellbore

Fig. 1.2 Multiphysical fields coupling in damage and fracture of rock

the chemical field affects the mechanical properties of rock (Rutqvist et al. 2002; Peirce and Siebrits 2005; Watanabe et al. 2012; Huang et al. 2015; Li and Laloui 2016; Jin and Zoback 2017).

1.2 Numerical Investigations for Damage and Fracture in Rock Considering Multiphysical Fields Coupling Theoretical and physical experimental solutions have certain limitations when they come to application in the analysis of damages and fractures in rocks considering multiphysical fields coupling; therefore, numerical simulation becomes the most

1.2 Numerical Investigations for Damage and Fracture in Rock …

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appropriate alternative. Numerical methods and models provide effective means of understanding and consequently controlling fracturing processes. Table 1.1 shows the advantages and limitations of conventional numerical methods for analysis of damage and fracture in rock. The finite element method (FEM) includes a systematic theory and techniques, extensive applicability, and multiphysical fields coupling; however, it derives solutions based on the unchangeable mesh, which makes its solutions near fracture tips inaccurate (Fu et al. 2011; Khoei et al. 2015). The extended finite element method (XFEM) uses enriched nodes and shape functions to simulate the discontinuum near fracture tips, but it is still challenging to solve the problems of multifield coupling in 3D (Elizaveta and Anthony 2013; Gordeliy and Peirce 2015; Haddad and Sepehrnoori 2015; Mohammadnejad and Andrade 2016). The discrete element method (DEM) can obtain discontinuous and subregional structures and rigid elements, but it has low computational efficiency, particularly with respect to fracture propagation along the element faces, and therefore is not suitable for continuum problems (Virgo et al. 2013; Peng et al. 2017). The finite element (FE)-discrete element (DE) method combines the characteristics of conventional FEM and DEM for continuum and discontinuum analysis; however, it still has low computational efficiency, particularly with respect to fracture propagation along the element faces (Munjiza et al. 2000; Owen et al. 2004; Lisjak et al. 2017). The mesh quality of the numerical method controls the accuracy and effectiveness of their solutions when it simulates the continuous damage evolution and fracture propagation behaviours in rock mass. A few high-performance adaptive algorithms have been proposed to automatically optimize the mesh based on the complexity of the problem (Babuška and Rheinboldt 1978; Zienkiewicz and Zhu 1992; Oden and Prudhomme 2001; Azadi and Khoei 2011). Figure 1.3 shows the basic flow of the high-performance adaptive FE algorithm: first, the Tolerance (error Table 1.1 Advantages and limitations of conventional numerical methods for damage and fracture of rock Method

Advantages

Limitations

Finite element method (FEM)

• Systematic theory and techniques • Extensive applicability • Multiphysical fields coupling

• Unchangeable mesh • Inaccurate solutions near fracture tips

Extended finite element method (XFEM)

• Enriched nodes and shape • Multiphysical fields coupling functions for simulating • Not suitable for 3D problems discontinuum near fracture tips

Discrete element method (DEM)

• Discontinuous and subregional • Low computational efficiency structures • Fracture propagation along element faces • Rigid elements • Not suitable for continuum problems

Finite element-discrete element method (FE-DE method)

• Combined characteristics of • Low computational efficiency FEM and DEM for continuum • Fracture propagation along and discontinuum analysis element faces

4

1 Introduction

Fig. 1.3 Basic process flow of the high-performance adaptive FE algorithms

limitation) is input and the FE solution is computed on the current mesh; then, the error estimation of the solution is made; if the solutions do not meet the error limitation, the mesh will be adaptively adjusted to form a new set of meshes; subsequently, the new meshes are used to recompute solutions until they satisfy the initial error limitation. Compared with the traditional method, the accuracy and efficiency of the adaptive FE algorithms are improved. It should be noted that the adaptive algorithms have some obvious advantages over traditional numerical methods with respect to solving challenging problems such as fracture, singularity, and eigenvalue problems.

1.3 Research Aims and Contents of the Book 1.3.1 Research Aims The research presented in this book has the following two objectives: (1) Continuous damage problem: the anisotropic continuous damage method and adaptive analysis of the wellbore stability of rocks. This work developed a method to investigate the continuous damaging in rocks and extended it to the anisotropic case. Using the continuous damage method, the wellbore stability analysis scheme, considering fluid, solid, and chemical coupling, was developed, and adaptive finite element analysis was used to solve the continuous damage problem. (2) Fracture propagation problem: multiscale propagation algorithms for fractures and adaptive analysis of fracturing in horizontal wells. This work developed the adaptive algorithm and numerical models for rock fracture analysis, developed the stimulated reservoir volume analysis scheme for horizontal wells, considering fluid, solid, and temperature coupling, and used adaptive finite element analysis to solve the fracture propagation problem.

1.3 Research Aims and Contents of the Book

5

Fig. 1.4 Structural chart of main contents and chapters

1.3.2 Research Contents The chapters in this book are mostly taken from published or written journal papers and are re-integrated into textual representations. Figure 1.4 presents a structural chart of the main contents and chapters: (1) Firstly, this Chapter introduces the research background and significances, numerical investigations, research aims and contents of this book. (2) In Chap. 2, FE algorithm for continuum damage evolution of rock considering hydro-mechanical (HM) coupling is introduced. (3) In Chap. 3, FE analysis for continuum damage evolution and wellbore stability of transversely isotropic rock considering hydro-mechanical coupling are introduced. (4) In Chap. 4, FE analysis for continuum damage evolution and inclined wellbore stability of transversely isotropic rock considering hydro-mechanical-chemical (HMC) coupling are introduced. (5) In Chap. 5, adaptive FE algorithm for damage detection of non-uniform EulerBernoulli beams with multiple cracks based on natural frequencies is introduced. (6) In Chap. 6, adaptive FE-DE analysis for the multistage hydrofracturing in naturally fractured reservoirs considering hydro-mechanical (HM) coupling is introduced. (7) In Chap. 7, adaptive FE-DE analysis for the multistage supercritical CO2 (SC-CO2 ) fracturing and microseismic modelling considering thermal-hydromechanical (THM) coupling is introduced. (8) In Chap. 8, adaptive FE-DE-FV algorithm for 3D multiscale propagation of hydraulic fracture network considering hydro-mechanical (HM) coupling is introduced. (9) Finally, Chap. 9 summarizes the main conclusions and outlooks of this book.

6

1 Introduction

References Abousleiman YN, Nguyen VX (2005) Poromechanics response of inclined wellbore geometry in fractured porous media. J Eng Mech 131(11):1170–1183 Azadi H, Khoei AR (2011) Numerical simulation of multiple crack growth in brittle materials with adaptive remeshing. Int J Numer Methods Eng 85(8):1017–1048 Babuška I, Rheinboldt WC (1978) A-Posteriori error estimates for the finite element method. Int J Numer Methods Eng 12(10):1597–1615 Bažant ZP, Salviato M, Chau VT, Visnawathan H, Zubelewicz A (2014) Why fracking works. J Appl Mech 81(10):1–10 Elizaveta G, Anthony P (2013) Coupling schemes for modeling hydraulic fracture propagation using the XFEM. Comput Meth Appl Mech Eng 253(1):305–322 Erarslan N, Williams DJ (2012) The damage mechanism of rock fatigue and its relationship to the fracture toughness of rocks. Int J Rock Mech Min Sci 56(56):15–26 Fu P, Johnson SM, Carrigan CR (2011) An explicitly coupled hydro-geomechanical model for simulating hydraulic fracturing in complex discrete fracture networks. Int J Numer Anal Meth Geomech 37(14):2278–2300 Gordeliy E, Peirce A (2015) Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems. Comput Meth Appl Mech Eng 283(283):474–502 Haddad M, Sepehrnoori K (2015) XFEM-based CZM for the simulation of 3D multiple-cluster hydraulic fracturing in quasi-brittle shale formations. Rock Mech Rock Eng 49(12):1–18 Huang Z, Winterfeld P, Xiong Y, Wu Y, Yao J (2015) Parallel simulation of fully-coupled thermalhydro-mechanical processes in CO2 leakage through fluid-driven fracture zones. Int J Greenhouse Gas Control 34:39–51 Jin L, Zoback MD (2017) Fully coupled nonlinear fluid flow and poroelasticity in arbitrarily fractured porous media: a hybrid-dimensional computational model. J Geophys Res: Solid Earth 122(10):7626–7658 Khoei AR, Hirmand M, Vahab M et al (2015) An enriched FEM technique for modeling hydraulically driven cohesive fracture propagation in impermeable media with frictional natural faults: Numerical and experimental investigations. Int J Numer Meth Eng 104(6):439–468 Krajcinovic D, Lemaitre J (1987) Continuum damage mechanics theory and applications. Springer, New York, USA Lee H, Chang C, Ong SH, Song I (2013) Effect of anisotropic borehole wall failures when estimating in situ stresses: a case study in the Nankai accretionary wedge. Mar Pet Geol 48:411–422 Lee H, Ong SH, Azeemuddin M, Goodman H (2012) A wellbore stability model for formations with anisotropic rock strengths. J Pet Sci Eng 96:109–119 Li C, Laloui L (2016) Coupled multiphase thermo-hydro-mechanical analysis of supercritical CO2 injection: benchmark for the In Salah surface uplift problem. Int J Greenhouse Gas Control 51:394–408 Lisjak A, Kaifosh P, He L, Tatonea BSA, Mahabadia OK, Grasselli G (2017) A 2D, fully-coupled, hydro-mechanical, FDEM formulation for modelling fracturing processes in discontinuous, porous rock masses. Comput Geotech 81:1–18 Marc-André B, Yan M, Stead D (2009) The role of tectonic damage and brittle rock fracture in the development of large rock slope failures. Geomorphol 103(1):30–49 Mohammadnejad T, Andrade JE (2016) Numerical modeling of hydraulic fracture propagation, closure and reopening using XFEM with application to in-situ stress estimation. Int J Numer Analy Methods Geomech 40(15):2033–2060 Mortara G (2010) A yield criterion for isotropic and cross-anisotropic cohesive-frictional materials. Int J Numer Anal Met 34(9):953–977 Munjiza A, Latham JP, Andrews KRF (2000) Detonation gas model for combined finite–discrete element simulation of fracture and fragmentation. Int J Numer Methods Eng 49(12):1495–1520

References

7

Nguyen VX, Abousleiman YN (2009) Naturally fractured reservoir three-dimensional analytical modeling: theory and case study. SPE Annual Technical Conference and Exhibition. Soc Pet Eng SPE-123900-MS Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comput Math Appl 41(5):735–756 Owen DRJ, Feng YT, de Souza Neto EA, Cottrell MG, Wang F, Andrade Pires FM, Yu J (2004) The modelling of multi-fracturing solids and particulate media. Int J Numer Meth Eng 60(1):317–339 Peirce AP, Siebrits E (2005) A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems. Int J Numer Methods Eng 63(13):1797–1823 Peng P, Ju Y, Wang Y, Wang S, Gao F (2017) Numerical analysis of the effect of natural microcracks on the supercritical CO2 fracturing crack network of shale rock based on bonded particle models. Int J Numer Anal Meth Geomech 41(18):1992–2013 Rutqvist J, Wu Y, Tsang C, Bodvarsson G (2002) A modeling approach for analysis of coupled multiphase fluid flow, heat transfer, and deformation in fractured porous rock. Int J Rock Mech Min Sci 39(4):429–442 Virgo S, Abe S, Urai JL (2013) Extension fracture propagation in rocks with veins: insight into the crack-seal process using Discrete Element Method modeling: DEM fracture-vein interaction. J Geophys Res Solid Earth 118(10):5236–5251 Watanabe N, Wang W, Taron J, Gorke UJ, Kolditz O (2012) Lower-dimensional interface elements with local enrichment: application to coupled hydro-mechanical problems in discretely fractured porous media. Int J Numer Methods Eng 90(8):1010–1034 Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput Meth Appl Mech Eng 101(1):207–224

Chapter 2

Finite Element Algorithm for Continuum Damage Evolution of Rock Considering Hydro-Mechanical Coupling

2.1 Introduction The finite element (FE) analysis technology presented in this chapter takes rock as the research object, a typical porous medium with heterogeneous material property, dispersedly distributed damage region under applied load and in situ stress, inorganic and organic pores to form the complex structure (Zhuang et al. 2015). Heterogeneous distribution and degeneration of material property (i.e., Young’s modulus and permeability) or strength property (i.e., compressive and tensile strength) make the assumption of homogeneity and continuity is not consistent with the actual situation; for some extreme cases, the state-of-the-art strength theory of rock based on continuum mechanics also confronts with severe problems challenging the reseachers’ best knowledge (Tien and Kuo 2001). Some technology, such as heterogeneous Young’s modulus simulation to describe the heterogeneity of rock, is developed (Zhu and Bruhns 2008), and the author of this book also has adopted this technology to form the heterogeneous rock with heterogeneous Young’s modulus and strength (Wang et al. 2016). The technology could be developed; the permeability is heterogeneous and has some relationship with the heterogeneity of Young’s modulus to form a new technology to describe the heterogeneity of rock in this chapter. Unconventional rock, such as shale, has the characteristic of low porosity and permeability; particularly the key technology of hydraulic fracking could create artificial fracture network with centralized distribution to increase the porosity and permeability for large-scale extraction (Bazant et al. 2014). The effective stress of porous medium will change with the fluid flow, pressure diffusion in pores and solid deformation correspondingly; in other words, hydro-mechanical (HM) coupling makes the response of the rock reflect complex time-dependent effect obviously. Also, for research, the single-porosity elastic model and dual-porosity elastic model (Wu et al. 2010) are developed successively; additionally, FE analysis technology for a new constitutive model considering the adsorption is putted forward recently (Zhang et al. 2008).

© Science Press 2021 Y. Wang, Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling, https://doi.org/10.1007/978-981-15-7197-8_2

9

10

2 Finite Element Algorithm for Continuum Damage Evolution …

For some conventional and simple problems, such as possessing regular solving domain with homogeneous material property, analytical method via the poroelastic theory has been proposed to solve a number of multiphysical fields coupling, wellbore stability and fracture problems (Cui et al. 1997; Nguyen and Abousleiman 2009), however, which unfortunately does not consider heterogeneity, plastic deformation and damage evolution and hence cannot serve as a practical and general method. Therefore, the author of this book was motivated to probe into this numerical area of rock fracture and damage evolution problems. The discrete fracture network (DFN) model (Warpinski et al. 2009) based on the fracture mechanics reflects both effective and efficient solving ability for single physical field and regular fracture system; belonging to this model, a typical extended finite element method (XFEM) has been successfully developed. Some researchers have made a series of achievements on the XFEM theory and algorithm (Zhuang et al. 2014), i.e., the problems of crack growth in pipes (Zhuang and Cheng 2011), multi-scales computation (Zhuang and Magfereti 2012), two-phase flows (Liao and Zhuang 2012), error estimation (Lin and Zhuang 2014), cracks in shells (Zeng et al. 2014) and crack crossing (Xu et al. 2014); additionally, the XFEM has been improved and applied to numerical simulation for hydraulic fracturing of shale rock in recent study (Wang et al. 2014). On the other hand, some researchers proposed the continuous model based on the damage mechanics to analyze the failure process, compared to above DFN model, which more easily handles the multiphysical fields coupling and complex medium containing pores and fractures problems to be consistent with the material behavior (Krajcinovic and Lemaitre 1987). Inspired from which, some numerical methods and simulation technologies based on damage analysis are proposed (Zhu and Bruhns 2008; Lu et al. 2013). The damage variable describing the compressive and tensile behaviors of rock was developed by mesoscopic damage mechanics (Zhu and Bruhns 2008); a novel, simple FE analysis technology of seepage in rock has been presented yet by author of this book with the further development of the continuous model and continuum damage variable (Wang et al. 2016). In this chapter, the continuum damage variable was developed by introducing a correction factor to consider the part of the bearing capacity of damage rock effectively. The well-known Biot constitutive theory (Biot 1941, 1954) is introduced firstly as the basis of proposed method; thereafter, solid and gas seepage equations coupled by the term of pore pressure are established to form the HM coupling. The FE analysis technology in this chapter achieves the results for coupled HM model of rock based on continuum damage evolution by implementing the following three-step strategy. In the first step, for establishing physical HM coupling model, the heterogeneous rock composes of Young’s modulus and permeability on each FE node obeying the Weibull distribution. Then FE solutions of HM model, such as effective stresses, permeability and porosity, etc., are obtained by a developed FE procedure in the second step. In the last step, the above effective stress solutions are used to calculate the damage variable on each FE node, and furthermore the Young’s modulus and permeability with damage on each FE node could be calculated respectively. The Young’s modulus and permeability with damage under the current load would be used to form a new heterogeneous physical model, and the procedure returns to the

2.1 Introduction

11

second step until the load steps progressively complete. This yields a simple, efficient, reliable and practical FE analysis technology that is able to make rock modeling and damage analysis in mining and petroleum engineering applications. The numerical examples presented later have shown that utilizing petrophysical heterogeneity simulation is well effective and feasible; in addition to the reliability, the results also show that the method is almost consistent with damage evolution under different load conditions, considering HM coupling is necessary for effective stress analysis.

2.2 Assignment of Petrophysical Heterogeneity Heterogeneity of rock is the research focus with some simulated difficulties currently; taking simplifications of homogenizing treatment, there will get obviously deviated results to actual situation. Various mineral particles and cementing materials constitute the common rock, in which forming process is extremely complex. Dispersedly distributed pores and microcosmic cracks in the rock exhibit inhomogeneity and heterogeneity of physical and mechanical properties. Petrophysical heterogeneity analysis widely adopts the Weibull distribution (Weibull 1951) to research the structural strength, fatigue and other problems of rock, which have obtained some satisfactory numerical simulation effect and the good consistency with the experiments (Zhu et al. 2006; Zhu and Bruhns 2008; Mahabadi et al. 2014). In order to characterize the heterogeneity of rock, in this study, the model is discrete as many elements by FEM, and the mechanical properties of each node on FE mesh are assumed to conform to a given Weibull distribution as defined in the following probability density function:   m−1 f (x, η, m) =

m η

0,

x η

  m  ,x ≥0 exp − ηx x |σ3 |, η takes the compressive strength C; on the other hand, when the microunit is in the dominant state of tension as σ3 ≤ σ2 ≤ σ1 ≤ 0 or σ1 ≥ 0, σ3 ≤ 0 and |σ3 | ≥ |σ1 |, η takes the tensile strength T. Here σ1 , σ2 , σ3 are three principal stresses, in which sign convention used throughout in this chapter is that compressive stress and strain are positive. As shown in Eq. (2.6), the damage variable D is associated with both the rock strength and the stress state under the current load. It is noted that the damage variable D of micro-unit keeps the original value in the unloading case. For rock solid damage under load, the Young’s modulus E˜ with damage is conducted as  √ m a I1 + J2

(2.7) E = E 0 (1 − D) = d E 0 exp − η The permeability of rock will enhance gradually due to the macroscopic damage generation and expansion; based on the relationship between permeability and strain, an effective technology of permeability damage evolution is proposed (Zhu and Bruhns 2008; Wang et al. 2016) to reflect the damage characteristic as



3  3  α p0 p0 φ 1 k˜ + S− 1+ = exp(γ D) = exp(γ D) k0 φ0 1+S Ks φ0 Ks (2.8) here k˜ is permeability with damage; k0 and φ0 are initial permeability and porosity respectively; γ is the permeability damage coefficient; and exp(γ D) is the continuum damage item to describe the fact that the permeability is enhanced by damage. Through some numerical examples test including the results given below, the Eqs. (2.7) and (2.8) describing the damage of rock reveal excellent effect.

2.5 Finite Element Analysis Strategy

15

2.5 Finite Element Analysis Strategy The finite element analysis (FEA) in this chapter achieves the results for coupled HM model of rock based on continuum damage evolution by implementing the following three-step strategy. (1) Heterogeneous model. The parameters η and m in Weibull distribution are given by user according to the heterogeneous property of rock; moreover the FE model of rock established for generating the FE mesh with each node possesses the different value for Young’s modulus E and permeability k to derive the heterogeneous rock, as described in Sect. 2.2. (2) FE solution. On the current load step, the effective stress σi j and gas pore pressure p could be calculated using Eqs. (2.2) and (2.4) to obtain solutions of HM model by the standard FE analysis, described in Sect. 2.3. (3) Damage analysis. The above effective stress σi j is used to calculate the damage variable D on each node by Eq. (2.6), and furthermore the Young’s modulus E˜ and permeability k˜ with damage on each node could be calculated by Eqs. (2.7) and (2.8) respectively, forming a new FE model, as described in Sect. 2.4. Then the procedure returns to the second step (i.e., FE solution) until the load steps progressively complete. The above three steps constitute a round of continuum damage analysis for rock with HM coupling, by which the gas seepage flow and rock deformation results could be obtained. The numerical examples below show the proposed FE analysis strategy is effective, reliable and effective.

2.6 Results and Discussion The proposed analysis strategy has been coded into a MATLAB program and partly used the FE solver of COMSOL Multiphysics to obtain solutions of governing equations with HM coupling. This section presents three interrelated numerical examples showing the excellent performance of the procedure. Throughout, the program is run on a DELL Optiplex 380 Intel (R) Core (TM) 2.93 GHz desktop computer. The first example is chosen to discuss the appropriate parameter selection of Weibull distribution to describe the heterogeneity for rock, so that the rock model researched below with heterogeneous Young’s modulus and permeability will be established reliably. The second example analyzes two damage evolution cases of heterogeneous rock with an inner circular hole under different load conditions, and the effective stress results of one typical model will be specially researched as the third example: considering HM coupling is compared with the single solid model without considering rock as pore structure. In last two examples, the triangle element is used and the shape parameter m of Weibull distribution for heterogeneous material property and strength property are set to be same throughout.

16

2 Finite Element Algorithm for Continuum Damage Evolution …

2.6.1 Finite Element Model of Heterogeneous Rock Considering plane strain model in Fig. 2.1, the geometric domain is square possessing an inner circular hole with a = 2 m, b = 0.127 m, c = 1 m here a is side length of the square; b is diameter of the hole; c is the thickness parameter. The next section discusses the influence of Weibull distribution parameters η and m on petrophysical heterogeneity. Take η = 1 and m = 2, 4, 6 as examples, the distribution results are obtained by the proposed method as Fig. 2.2. The distributed values in the entire domain are from 0.11 to 2.24, from 0.21 to 1.51 and from 0.38 to 1.34 in Fig. 2.2a–c respectively, in which it can be seen that the distributed values are getting closer together to the mean value η with homogeneity index parameter m taking bigger value. These results are well consistent with the Weibull distribution theory as introduced in Sect. 2.2. Considering the above influence of Weibull distribution parameters, m = 5 is adapted to simulate the heterogeneous material parameter as a moderate distribution way to let values be neither too big nor too small, and η = E 0 = 16.32 GPa is adapted as the initial, mean value of Young’s modulus. As shown in Fig. 2.3, the distributed result of Young’s modulus E is generated, and the distributed results of permeability k could be obtained according to inversely proportional relationship with E as Fig. 2.4, and here mean value of rock permeability k 0 is adapted as 1.9 × 102 mD (1 mD = 10-3 µm2 ). Then the initial FE model of heterogeneous rock is established successfully by the proposed method in this chapter. Fig. 2.1 Geometric model

2.6 Results and Discussion

Fig. 2.2 Simulation of heterogeneous distribution Fig. 2.3 Heterogeneous distribution of initial Young’s modulus

17

18

2 Finite Element Algorithm for Continuum Damage Evolution …

Fig. 2.4 Heterogeneous distribution of initial permeability

2.6.2 Rock Damage Analysis Under Different Conditions Considering the established FE model of heterogeneous rock in Example 1 with physical parameters as in Table 2.1, some of these parameters are chosen from physical properties of rock in references (Detournay and Cheng 1993; Zhang et al. 2008). Two different load conditions for damage evolution are analyzed comparatively: the Model I in Fig. 2.5 has uniform tensile load P on upper and lower edges, and the Model II in Fig. 2.6 has uniform pressure load P in inner circular hole and simply supports boundary outside. The computed damage evolution results of two different load conditions as load P from 0 to 50 MPa are shown in Figs. 2.7 and 2.8. The region near the hole is selected because the damage variable D has dramaticly changed there. As shown in Figs. 2.7 and 2.8, the damage regions expand discontinuously with P increasing as the heterogeneity of rock. In Fig. 2.7, the damage region near the hole expands Table 2.1 Physical parameters of model

Parameter

Value

Young’s modulus, E/GPa

16.32

Poisson’s ratio, ν

0.33

Biot coefficient, α

0.79

Druker-Prager constant, a   Gas density, ρa / kg/m3

0.29 0.717

Damage factor, d

1.0

Permeability damage coefficient, γ

0.1

Initial porosity, φ0

0.019

Initial permeability, k0 /mD

1.9 × 102

Motion viscosity coefficient, μ/(Pa s)

1.84 × 10–5

Compressive strength, C/MPa

103

Tensile strength, T /MPa

7.5

2.6 Results and Discussion

19

Fig. 2.5 Model I with outer tensile load

Fig. 2.6 Model II with inner compressive load

gradually to the horizontal direction of the model by stress consideration; while in Fig. 2.8, the damage region around the hole expands outward consistently without significant damage propagation. These above results are consistent with damage theory and regular experiments to evaluate the effectiveness of rock damage analysis of the proposed method.

2.6.3 Effective Stress Analysis by Hydro-Mechanical Coupling Influence The effective stress characteristic of rock HM coupling will be analyzed by the proposed method, utilizing the plane strain Model II in Fig. 2.6 widely used as

20

2 Finite Element Algorithm for Continuum Damage Evolution …

Fig. 2.7 Damage variable D of model I with outer tensile load

wellbore model in mining and petroleum engineering. For comparative analysis, the stress result of single solid model was computed, which had the same geometry and physical parameters, considering petrophysical heterogeneity and damage analysis as HM coupling case in Fig. 2.6. The radial and tangential stress results at cross section A-A in Fig. 2.6 are analyzed as P = 50 MPa in Figs. 2.9 and 2.10 respectively, where R is the radius of the wellbore and r is the distance from the wellbore center. As shown in Fig. 2.9, it can be seen that both radial stress of HM model and single solid model are almost the same at near-field and far-filed of wellbore, but the stress result at the middle-filed of HM model is bigger than that of the solid model. The tangential stress is a decisive factor for wellbore stability, as shown in Fig. 2.10, the result of single solid model at near-field is significantly smaller than HM coupling’s, and then the wellbore instability would tend to happen in single solid model case. Therefore, multiphysical fields coupling model and the effective stress analysis technology are significant and should be developed reasonably.

2.6 Results and Discussion

Fig. 2.8 Damage variable D of model II with inner compressive load Fig. 2.9 Radial stress results of two models

21

22

2 Finite Element Algorithm for Continuum Damage Evolution …

Fig. 2.10 Tangential stress results of two models

2.7 Conclusions HM model of heterogeneous rock based on continuum damage evolution and FE analysis technology with suitable analysis strategy for effective and reliable computation of the seepage and damage of rock have been presented. Comprehensive utilization of continuum damage theory with a number of other auxiliary techniques (including the Weibull distribution, classic Biot constitutive relation and DP strength criterion) has yielded a simple, efficient and reliable FE procedure that describes the damage behavior for being consistent with the property of rock to be achieved. The techniques integration removes major difficulties such as heterogeneity, multiphysical fields coupling and damage evolution in the solution of the seepage flow, collapse and fracture problems of rock, which enhances the overall efficiency and reliability of solution. Results for typical numerical examples, including ones known to be engineering problems, have shown that the present method is competitive and possesses the potential for further extension to more complex mechanic problems (e.g., anisotropy damage model and wellbore stability problems). A recent study of transversely isotropic hydro-mechanical-damage coupling model successfully solved wellbore stability, as well as rock instability problems, so paving the way for corresponding anisotropic damage tensor and failure problems of brittle material in complex conditions to be solved efficiently and accurately by applying the FE strategy of the proposed procedure. These advances will be reported in future researches as they are developed.

References Bazant ZP, Salviato M, Chau VT, Viswanathan H, Zubelewicz A (2014) Why fracking works. J Appl Mech 81(10):1–10 Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164

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Biot MA (1954) Theory of stress strain relations in anisotropic viscoelasticity and relaxation phenomena. J Appl Phys 25(11):1385–1391 Cui L, Cheng AHD, Abousleiman YN (1997) Poroelastic solution for an inclined borehole. J Appl Mech 64(1):32–38 Detournay E, Cheng AHD (1993) Fundamentals of Poroelasticicty. Anal Des Methods 2:113–171 Krajcinovic D, Lemaitre J (1987) Continuum damage mechanics theory and applications. SpringerVerlag, New York, USA Liao JH, Zhuang Z (2012) A consistent projection-based SUPG/PSPG XFEM for incompressible two-phase flows. Acta Mech Sin 28(5):1309–1322 Lin ZJ, Zhuang Z (2014) Enriched goal-oriented error estimation for fracture problems solved by continuum-based shell extended finite element method. Appl Math Mech 35(1):33–48 Lu YL, Elsworth D, Wang LG (2013) Microcrack-based coupled damage and flow modeling of fracturing evolution in permeable brittle rocks. Comput Geotech 49:226–244 Mahabadi OK, Tatone BSA, Grasselli G (2014) Influence of microscale heterogeneity and microstructure on the tensile behavior of crystalline rocks. J Geophys Res Solid Earth 119(7):5324–5341 Nguyen VX, Abousleiman YN (2009) Naturally fractured reservoir three-dimensional analytical modeling: theory and case study. SPE123900 Roberton EP, Christiansen RL (2005) Modeling permeability in coal using sorption-induced strain data. SPE97068 Tien YM, Kuo MC (2001) A failure criterion for transversely isotropic rocks. Int J Rock Mech Min Sci 38(3):399–412 Wang T, Gao Y, Liu ZL, Wang YH, Yang LF, Zhuang Z (2014) Numerical simulations of hydraulic fracturing in large objects using an extended finite element method. J Tsinghua Univ (Sci Tech) 10:1304–1309 Wang YL, Liu ZL, Lin SC, Zhuang Z (2016) Finite element analysis of seepage in rock based on continuum damage evolution. Eng Mech 33(11):29–37 Warpinski NR, Mayerhofer MJ, Vincent MC, Cipolla CL, Lolon EP (2009) Stimulating unconventional reservoirs: maximizing network growth while optimizing fracture conductivity. J Can Petrol Technol 48(10):39–51 Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18:293–297 Wu Y, Liu JS, Elsworth D, Chen Z, Connell L, Pan Z (2010) Dual poroelastic response of a coal seam to CO2 injection. Int J Greenhouse Gas Control 4(4):668–678 Xu DD, Liu ZL, Liu XM, Zeng QL, Zhuang Z (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Comput Mech 54(2):489–502 Zeng QL, Liu ZL, Xu DD, Zhuang Z (2014) Modeling stationary and moving cracks in shells by X-FEM with CB shell elements. Sci Chi Technol Sci 57(7):1276–1284 Zhang HB, Liu JS, Elsworth D (2008) How sorption-induced matrix deformation affects gas flow in coal seams A new FE model. Int J Rock Mech Min Sci 45(8):1226–1236 Zhu WC, Bruhns OT (2008) Simulating excavation damaged zone around a circular opening under hydromechanical conditions. Int J Rock Mech Min Sci 45(5):815–830 Zhu WC, Liu J, Yang TH, Sheng JC, Elsworth D (2006) Effects of local rock heterogeneities on the hydromechanics of fractured rocks using a digital-image-based technique. Int J Rock Mech Min Sci 43(8):1182–1199 Zhuang Z, Cheng BB (2011) A novel enriched CB shell element method for simulating arbitrary crack growth in pipes. Sci Chin Phys Mech Astron 54(8):1520–1531 Zhuang Z, Magfereti M (2012) The recent research progress in computation solid mechanics at multi-scales. Chin Sci Bull 57(36):4683–4688 Zhuang Z, Liu ZL, Cheng BB, Liao JH (2014) Extended finite element method. Elsevier/Tsinghua University Press, Amsterdam/Beijing, Netherlands/China Zhuang Z, Liu ZL, Wang YL (2015) Fundamental theory and key mechanical problems of shale oil gas effective extraction. Chin Q Mech 33(1):8–17

Chapter 3

Finite Element Analysis for Continuum Damage Evolution and Wellbore Stability of Transversely Isotropic Rock Considering Hydro-Mechanical Coupling

3.1 Introduction In recent years, the wellbore instability problems always appear in practical shale gas exploitation in China. Due to the anisotropic mechanical behaviors of unconventional shale and lacking theoretical basis, the mud weight of horizontal well drilling mainly depends on experience, which leads to high cost and serious environmental problems (Zhuang et al. 2015). In experiments research, anisotropic shale is fabric-related and results from partial alignment of plate-like minerals such as clay, whose parameters exhibit the typical transversely isotropic properties by testing the physical parameters (Lo et al. 1986; Abousleiman et al. 2010). In order to research the characteristics of shale in China, some researchers had done the X-Ray diffraction and scanning electron microscope experiments analysis and had obtained the results that the microstructure of ordinary and rich-organic shale was very compact and the organic matter and minerals had a certain direction; the material and strength parameters exhibited apparent heterogeneity and typical transversely isotropic characteristic (Yang et al. 2013). The stability problem of a wellbore drilled into a thinly laminated anisotropic rock formation in this chapter is shown in Fig. 3.1, which is assumed in transversely isotropic formation coordinate system (TCS) xt − yt − z t ; the plane parallel and direction vertical to bedding are frequently replaced below by the terminologies as isotropic plan and transverse direction respectively. Drilling a borehole with a given mud pressure P into a formation, which is fully saturated with a pore fluid and subjected to the preexisting in situ stresses (SH is maximum horizontal principal stress; Sh is minimum horizontal principal stress; Sv is vertical stress) initially in static equilibrium, disturbs the state of stress in the vicinity of borehole. The finite element analysis (FEA) technology presented in this chapter takes the laminated rock as the research object, a typical porous medium with transversely isotropic property and strength, dispersedly distributed damage region under applied mud pressure and in situ stresses, to obtain the collapse and fracture pressure as safe mud weight. Unconventional rock, such as shale, has the characteristic of low porosity and permeability; particularly the key technology of horizontal well drilling could create © Science Press 2021 Y. Wang, Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling, https://doi.org/10.1007/978-981-15-7197-8_3

25

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3 Finite Element Analysis for Continuum Damage Evolution …

Fig. 3.1 Wellbore in laminated rock formation subjected to in situ stresses

damage and fracture propagation around the wellbore to change the porosity and permeability (Bažant et al. 2014). The effective stress of porous medium will change with the fluid flow, pressure diffusion in pores and solid deformation correspondingly; in other words, hydro-mechanical (HM) coupling makes the response of the rock reflect complex time-dependent effect obviously. Also, for research, the singleporosity elastic model and dual-porosity elastic model (Wu et al. 2010) are developed successively; additionally, the FEA technology for a new constitutive model considering the adsorption is putted forward recently (Zhang et al. 2008). For some conventional and simple problems, such as possessing regular solving domain with homogeneous material property, analytical method via the poroelastic theory has been proposed to solve a number of multiphysical fields coupling, wellbore stability and fracture problems (Abousleiman and Nguyen 2005; Nguyen and Abousleiman 2009), and a related method for an inclined borehole in a special transversely isotropic formation has been developed (Hoang et al. 2009; Chen and Abousleiman 2016). By utilizing the poroelastic analytical solutions and introducing the isotropic rock strength criterion, a weak plane model for wellbore stability of anisotropic formations has been developed (Lu et al. 2012). However, the analytical method unfortunately does not consider heterogeneity, plastic deformation and damage evolution and hence cannot serve as a practical and general method. Therefore, some reseachers were motivated to probe into the numerical area of anisotropic rock fracture, damage analysis and wellbore stability problems. In numerical methods, wellbore stability problems of transversely isotropic shale were analyzed using finite element method (FEM) based on the anisotropic institutive relationship and strength criterion, but the damage and seepage were not considered for simplicity (Roberto et al. 2009). Some FEA technology adopted two kinds of rock materials with different strength to simulate the transversely isotropic characteristic, and obtained failure modes for different dip angles of formation (Liang et al. 2005). Rock is a typical brittle medium, damage and fracture occur under load (Xiao et al. 2015, 2016), i.e. in situ stresses and mud pressure in wellbore, so numerical methods for damage and fracture have been developed successively. The discrete fracture network (DFN) model based on the fracture mechanics reflects both effective and efficient solving ability for single physical field and regular fracture system

3.1 Introduction

27

(Warpinski et al. 2009); extended finite element method (XFEM) as one of the typical methods is successfully developed. Some researchers have made a series of achievements on the basic theory and algorithm of XFEM for complex fracture problems, i.e. crack growth in pipes, multi-scales computation, two-phase flows, error estimation, cracks in shells and crack crossing (Zhuang and Cheng 2011; Liao and Zhuang 2012; Lin and Zhuang 2014; Zeng et al. 2014; Xu et al. 2014); additionally, XFEM has been improved and applied to simulate the hydraulic fracturing of shale in recent study (Wang et al. 2014). On the other hand, some researchers proposed the continuous model based on the damage mechanics to analyze the failure process, compared to the above DFN model, which is easier to handle the multiphysical fields coupling and complex medium containing pores and fractures problems to be consistent with the material behavior (Krajcinovic and Lemaitre 1987). Inspired from which, some numerical methods and simulation technologies based on damage analysis are proposed (Zhu and Bruhns 2008; Lu et al. 2013). The damage variable describing the compressive and tensile behaviors of rock is developed by mesoscopic damage mechanics (Zhu and Bruhns 2008); a novel, simple FEA technology of seepage in rock has been presented yet by the author of this book with the further development of the continuous model and continuum damage variable (Wang et al. 2016). In this chapter, the continuum damage variable is developed effectively into a damage tensor introducing the stresses state under the current load and anisotropic strength to consider the bearing capacity; utilizing the damage tensor, the proposed method will develop finite element (FE) algorithm to obtain the stress solutions with damage, and apply to the wellbore stability analysis of transversely isotropic rock. Over the past several decades, many researchers have devoted considerable efforts to the study of rock anisotropy, from both the theoretical and the experimental points of view. Many scholars have investigated mechanical properties of both nature and synthetic transversely isotropic rocks under varied confining pressures. A more general criterion, expressed as a quadratic function for anisotropic materials, was proposed (Hill 1950). This criterion is an extension of von Mises isotropic criterion. While von Mises and Hill criteria assume that the strength of the material is independent of hydrostatic stress and is suitable for metals and composite materials, they may not be directly applicable to geological materials because the strength behavior of most geological materials is dependent on the hydrostatic stress. Pariseau (1968) extended Hill’s criterion to account for the effect of the hydrostatic stresses. Wellbore stability problems of transversely isotropic shale were analyzed using FEM based on the Pariseau strength criterion to analyze the failure in isotropic plane and transverse direction, and the collapse pressure was obtained successfully (Roberto et al. 2009), which was introduced in this chapter to check the failure. The well-known Biot constitutive theory (Biot 1941, 1954) is introduced and developed into transversely isotropic case firstly as the basis of the proposed method; thereafter, the FE formulation of solid and gas seepage equations coupled by the term of pore pressure are established. The FEA technology in this chapter achieves the results for hydro-mechanical-damage (HMD) coupling model of rock based on damage tensor by implementing the following three-step strategy. In the first step, for wellbore stability problems of laminated rock, a numerical three-dimensional (3D)

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3 Finite Element Analysis for Continuum Damage Evolution …

FE model is established with in situ stress boundary conditions (BCs) for declined borehole with different inclinations and azimuth angles; and then the FE solutions as effective stresses and permeability are obtained. The damage tensor, the stress and permeability with damage would be calculated in the second step. In the last step, by utilizing the Pariseau strength criterion, the pressure load of mud on borehole will increase until both of the collapse pressure (lower mud weight) and fracture pressure (upper mud weight) would be achieved when the procedure finishes. This yields a simple, efficient, reliable and practical FEA technology that is able to make transversely isotropic rock modeling and wellbore stability analysis in petroleum engineering applications. The numerical examples presented later have shown that utilizing the FE model of the inclined wellbore, damage analysis and Pariseau strength criterion is well effective and feasible. In addition to the reliability, the numerical results show that the method is almost consistent with the theoretical analysis; considering transversely isotropic characteristic of laminated rock for the failure regions analysis, collapse and fracture pressure computation is necessary.

3.2 Finite Element Analysis Strategy The FEA technology in this chapter achieves the results for wellbore stability analysis of transversely isotropic rock with HMD coupling by implementing the three-step FEA strategy below. (1) FE solution. For the wellbore stability problems of the laminated rock, the numerical 3D FE model will be established with in situ stress boundary conditions (BCs) for declined borehole with different inclinations and azimuth angles. Under the current pressure load of mud on borehole, the effective stress σet and permeability kt could be calculated to obtain solutions by the standard FEA technology, as described in Sect. 3.3. (2) Damage tensor calculation. The aforementioned effective stress σet and the compressive or tensile strength are used to calculate the damage tensor Dt , and furthermore the stress σ˜ t and permeability kt with damage could be calculated respectively, by which the damage state of the rock under the current load could be described in Sect. 3.4. (3) Wellbore stability analysis. By utilizing the Pariseau strength criterion, the stress σ˜ t with damage will be used to check if the compressive or tensile situation happens in the isotropic plane and transverse direction of laminated rock as described in Sect. 3.5. The pressure load of mud on borehole will increase until both of the collapse pressure (lower mud weight) and fracture pressure (upper mud weight) are achieved; otherwise the procedure returns to the first two progressively. The aforementioned three steps constitute a round of FE solving, damage tensor calculation and wellbore stability analysis, by which the instability mode, collapse

3.2 Finite Element Analysis Strategy

29

and fracture pressure of transversely isotropic rock could be obtained to exhibit that there would be obviously different characteristics with isotropic case. The numerical examples below show that the proposed FEA strategy is effective, reliable and effective.

3.3 Finite Element Solution 3.3.1 Constitutive Equation The rock reservoir has the typical nonlinear characteristics plastic deformation and damage evolution; however, the nonlinear model will increase the difficulty and calculation cost for analysis, so the linear elastic models could be used in most of the problems. The proposed method in this chapter takes the hypothesis of linear elasticity and small deformation, and introduces the damage analysis in the study to describe the nonlinear behavior below. On the other hand, the Biot constitutive is a well established institutive for elastic pore medium (Detourmay and Cheng 1993); using it a coupled HM model will be derived from the proposed method. The anisotropic Biot constitutive equation (Cheng 1997) can be expressed as σ e = σ − α p = Cε − α p

(3.1)

T  where the σe is stress vector as σxet σ yet σzet τxet yt τ yet zt τxet zt ; the compressive stress  T is positive throughout this chapter; ε is strain vector as εxt ε yt εzt γxt yt γ yt zt γxt zt ; α is Biot coefficient vector as { αx α y αz 0 0 0 } T ; in transversely isotropic case, αx and α y are equal parameters in isotropic plane expressed as αh and αz is a parameter in transverse direction expressed as αv in the following content; p is the pore pressure; C is the stiffness matrix. For general anisotropy, the constitutive relation contains 28 independent material coefficients. For materials with three mutually orthogonal planes of elastic symmetry, known as orthotropy, there exist 13 independent material coefficients; furthermore, stiffness matrix C of transversely isotropic case could be simplified as ⎡

M11 M12 M13 ⎢ M11 M13 ⎢ ⎢ M33 ⎢ C=⎢ ⎢ ⎢ ⎣ sym.

0 0 0 M44

0 0 0 0 M55

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ M55

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3 Finite Element Analysis for Continuum Damage Evolution …

in which the drained elastic modulus can be expressed in terms of engineering constants as



E h E v − E h νv2 E h E v νh − E h νv2

, M12 =

, M11 = (1 + νh ) E v − E v νh − 2E h νv2 (1 + νh ) E v − E v νh − 2E h νv2 M13 =

E h E v νv E v2 (1 − νh ) Eh , , M = , M44 = 33 2 2 E v − E v νh − 2E h νv E v − E v νh − 2E h νv 2(1 + νh )

M55 = G v where E h and νh are drained Young’s modulus and Poisson’s ratio in the isotropic plane; E v and νv are similar quantities related to the direction of the axis of symmetry (transverse direction) and G v is the shear modulus related to the direction of the axis of symmetry. The principal direction of permeability is orthotropic with principal axes assumed to be coinciding with the elastic ones, so the permeability tensor is diagonal as ⎤ k xt k xt yt k xt zt kt = ⎣ k yt k xt zt ⎦ sym. kzt ⎡

where k xt and k yt have the equally initial permeability kh in isotropic plane; k zt has the initial permeability kv in transverse direction; as the non-diagonal parameters, k xt yt and k xt zt have the initial permeability kh ; k yt zt has the initial permeability kv .

3.3.2 Finite Element Formulation With aforementioned Biot constitutive equation, the solid and seepage control equations can be discretized utilizing FEM as Ku = P

(3.2a)



Ht¯ + S pt¯i−1 +t¯ = Spt¯i−1 + F

(3.2b)

where K is the stiffness matrix; u is the displacement vector on the FE nodes and P is the load vector; H is the conduction matrix; S is the memory matrix; F is the fluid convergence vector; p is the pore pressure; t¯ is time; t¯ is the time step; pt¯ i−1 is the pore pressure at t¯i−1 ; pt¯ +t¯ is the pore pressure at t¯i−1 + t¯. The load P i−1 in Eq. (3.2a) considers the pore pressure p as coupling relationship, whose pore pressure p is iterative FE solution in Eq. (3.2b). In the proposed method, the FE

3.3 Finite Element Solution

31

steady solver of COMSOL Multiphics (COMSOL Inc. 2010) is taken to obtain the long-time solutions.

3.3.3 Finite Element Model and Adaptive Mesh Refinement for Wellbore Stability Analysis The process of developing the anisotropic wellbore stability model involves multiple stress transformations between the different reference coordinate systems. Four reference coordinate systems are defined: global coordinate system (GCS), in situ stress coordinate system (ICS), borehole coordinate system (BCS), and transversely isotropic formation coordinate system (TCS, as mentioned in Fig. 3.1). Each of these and their mutual relationships are shown in Fig. 3.2. Based on the coordinate systems and their relationships defined, we obtained rotation matrices that are needed to transform stress components from one reference coordinate system to the other. Figure 3.2a shows the relationship between ICS with the axes xs − ys − z s and GCS with the axes X e − Ye − Z e . The GCS is defined with X e in the north, Ye in the east, and Z e pointing vertically down. The ICS is defined with xs in the minimum horizontal principal stress Sh , ys in the maximum horizontal principal stress SH , and z s in the overburden Sv . Their relationship is defined by the two angles, azimuth αs and deviation angle βs . It is typically assumed in many geomechanical studies that the overburden is acting in the vertical direction, but in reality it may not be vertical particularly in fields where the surface topology is significantly changed, or the subsurface geological structures such as folds and faults are complex. To take the most general case into account, we introduce the angle βs which is measured between axis Z e and axis z s . The rotation matrix for the transformation of stress components from ICS to GCS was derived by initially aligning the axes xs − ys − z s of ICS with the axes X e − Ye − Z e of GCS, and then applying two separate rotations around the selected axes. The first rotation about the axis z s rotates the axes xs − ys − z s counter- clockwise by an angle of αs . The resulting new frame would be referred to

Fig. 3.2 Relationship between the coordinate systems

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3 Finite Element Analysis for Continuum Damage Evolution …

as the axes I − J − K . The second step rotates the axes I − J − K about the axis J by a counter- clockwise angle of βs producing the final frame as shown in Fig. 3.2a. These two rotations correspond with the rotation matrix T s expressed in Eq. (3.3a), representing the transformation of stress components from ICS to GCS. Figure 3.2b shows the relationship between BCS with the axes xb − yb − z b and GCS with the axes X e −Ye −Z e . The configuration of a borehole with respect to GCS is defined by its azimuth αb measured counter-clockwise from axis X e (North direction) to axis xb aligned with the top of borehole, and deviation angle βb measured between axis Z e (down direction) and the axis z b (borehole axis). The transformation matrix Tb of stress components between GCS and BCS is derived by the same approach as Eq. (3.3a) and is expressed by Eq. (3.3b). Figure 3.2c shows the relationship between TCS with the axes xt − yt − z t and GCS with the axes X e − Ye − Z e , in which transversely isotropic formation with isotropic plane xt − yt and transverse direction aligned along axis z t . Transversely isotropic formation has the dip angle defined by γt measured between axis xt (reverse direction of axis xt ) and the projection of axis xt on the horizontal plane (X e − Ye ), and the dip direction defined by ϕt measured between axis X e and the projection of the axis xt on the horizontal plane. So the configuration of deviation angle and azimuth of the axes xt − yt − z t with respect to GCS could be defined as βt (= γt ) and αt (= φt − 180◦ ) respectively. The rotation matrix for the transformation of stress components from GCS to TCS is derived by the same approach as used to derive Eq. (3.3a). Utilizing the deviation angle βt and azimuth αt , the rotation matrix Tt for transforming stress components in GCS to those in WCS is expressed by Eq. (3.3b). ⎡

⎤⎡ ⎤ cos βs 0 sin βs cos αs sin αs 0 Ts = ⎣ 0 1 0 ⎦⎣ sin αs cos αs 0 ⎦ − sin βs 0 cos βs 0 0 1 ⎡ ⎤ cos αs cos βs sin αs cos βs sin βs = ⎣ − sin αs cos αs 0 ⎦ − cos αs sin βs − sin αs sin βs cos βs ⎡ ⎤⎡ ⎤ cos βb 0 sin βb cos αb sin αb 0 Tb = ⎣ 0 1 0 ⎦⎣ sin αb cos αb 0 ⎦ − sin βb 0 cos βb 0 0 1 ⎡ ⎤ cos αb cos βb sin αb cos βb sin βb = ⎣ − sin αb cos αb 0 ⎦ ⎡

Tt0

− cos αb sin βb − sin αb sin βb cos βb

⎤⎡ cos γt 0 sin γt cos γt ⎢ ⎥⎢ =⎣ 0 1 0 ⎦⎣ 0 − sin γt 0 cos γt − sin γt ⎡ sin φt cos γt cos φt cos γt ⎢ = ⎣ − cos φt sin φt − sin φt sin γt − cos φt sin γt

(3.3a)

(3.3b)

⎤⎡ ⎤ 0 sin γt cos(φt − 180◦ ) sin(φt − 180◦ ) 0 ⎥⎢ ⎥ 1 0 ⎦⎣ sin(φt − 180◦ ) cos(φt − 180◦ ) 0 ⎦ 0 0 1 0 cos γt ⎤ sin γt ⎥ (3.3c) 0 ⎦ cos γt

3.3 Finite Element Solution

33

The rotation matrices obtained can be used to transform stress components from a reference coordinate system to another. In situ principal stresses in ICS can be transformed to the different stress components in BCS using Eq. (3.4a); the stresses in BCS are projected into TCS by using Eq. (3.4b).

T

SBCS = TbT Ts SICS TbT Ts

(3.4a)

T

σTCS = TtT Tb σBCS TtT Tb

(3.4b)

in which ⎡

SICS

⎡ ⎤ ⎤ Sxb Sxb yb Sxb zb 0 0 =⎣ SH 0 ⎦, SBCS = ⎣ S yb S yb zb ⎦ sym. Sv sym. Szb Sh

where superscript T is the transpose of matrix; σTCS and σBCS are stresses in TCS and BCS respectively. In order to deal with all the in situ stress boundary conditions (BCs) flexibly, the 3D FE model is established in BCS as shown in Fig. 3.3. To reduce the stress concentration effect of in situ stress application, each side length a of the 3D FE model should be at least ten times bigger than the diameter d of the wellbore. According to the established FE model, the stress state of plane at the middle section of the wellbore as Fig. 3.3 is consistent with the practical in situ stress, whose technology is suitable by verification of numerical examples containing the examples below, so taking this stress on the plane as the stress state with current well deviation angle and azimuth is appropriate. In the aforementioned 3D FE model, the different types of stress concentration around wellbore will appear in different in situ stress BCs and pressure in borehole, so the mesh at near-field of wellbore should be denser than far-filed to obtain more accurate solutions. On the other hand, too dense mesh would cause overmuch solving

Fig. 3.3 FE model and stability analysis plane of inclined wellbore

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3 Finite Element Analysis for Continuum Damage Evolution …

Fig. 3.4 Adaptive mesh refinement for FE model of inclined wellbore

time; on the contrary, sometimes denser mesh cannot get more accurate solutions. The meshing method adapted here is adaptive mesh refinement method, by which the user gives an initial coarse FE mesh and specified error tolerance Tol, and then the computed results on the final adaptive FE mesh will satisfy error tolerance Tol. This adaptive method has some advantages compared with the traditional FEM, which possesses high precision and high efficiency, and suits for different in situ stress BCs and loading conditions. Figure 3.4 is an adaptive mesh for the FE model of the inclined wellbore by the proposed method, initial coarse FE mesh is given, and final adaptive FE mesh is obtained as Tol = 5% for controlling the stress solution. From Fig. 3.4, it can be seen that the mesh is become denser automatically in final adaptive mesh than the initial coarse FE mesh; around the wellbore, optimally distributed elements are adopted to handle the problem of stress concentration. In order to characterize the heterogeneity of rock, in the preliminary study of the author, the model is discrete as many elements by FEM, and the mechanical properties of each node on the FE mesh are assumed to conform to a given Weibull distribution as defined in the following probability density function (Wang et al. 2016):   m−1 f (x, η, m) =

m η

0,

x η

  m  ,x ≥0 exp − ηx x |σ3 |, η takes the compressive strength C; on the other hand, when the micro-unit is in the dominant state of tension as σ3 ≤ σ2 ≤ σ1 ≤ 0 or σ1 ≥ 0,σ3 ≤ 0 and |σ3 | ≥ |σ1 |, η takes the tensile strength T . Here σ1 , σ2 , σ3 are three principal stresses, in which sign convention used throughout this chapter is that compressive stress and strain are positive. As shown in Eq. (3.6), the damage variable D is associated with both the rock strength and the stress state under the current load. For the transversely isotropic medium, the damage may occur in each principle direction depending on the stress state under the current load and the anisotropic

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3 Finite Element Analysis for Continuum Damage Evolution …

strength property. In this chapter, the uniaxial tensile, compressive and shear strength are introduced and damage variable developed into the damage tensor is: ⎡

⎤ ⎡ ⎤ Dxt Dxt yt Dxt zt g(ηha ) g(ηhs ) g(ηvs ) Dt = ⎣ g(ηha ) g(ηvs ) ⎦ D yt D yt zt ⎦ = ⎣ sym. g(ηva ) sym. Dz t

(3.7)

where g(ηi ) is the damage function as Eq. (3.6); ηha and ηva are strength parameters in isotropic plane and transverse direction respectively; they adopt the uniaxial compressive strength Ch and Cv as the medium is compressive (0 ≤ σi , i = 1, 2, 3) in principle direction, and vice versa, the uniaxial tensile strength as Th and Tv would be adapted; ηhs and ηvs are the shear strength in transverse direction, and they adopt the 0.8-1.0 times the tensile strength (Jaeger et al. 2009). All these aforementioned strength parameters can be tested by traditional rock mechanical experiments.

3.4.2 Stress and Permeability with Damage In the transversely isotropic formation coordinate system (TCS), the solutions of effective stress σte and permeability kt have been computed by FEM in Sect. 3.3. Considering the effect of damage, the effective stress should be reduced by the damage tensor according to the strain equivalence principle (Krajcinovic 2000); the permeability of rock will enhance gradually due to the macroscopic damage generation and expansion; based on the relationship between permeability and strain (Zhu and Bruhns 2008; Wang et al. 2016), a reliable technology of effective stress and permeability damage evolution is proposed to reflect the damage characteristic as σ˜ t = σte /(1 − Dt )

(3.8a)

kt = kt × exp(γ Dt )(φ/φ0 )3

(3.8b)

here effective stress σte is expressed as tensor form: ⎤ σxet τxet yt τxet zt ⎥ ⎢ σte = ⎣ σ yet τ yet zt ⎦ sym. σzet ⎡

where σ˜ t is the stress with damage; 1−Dt is the damage item to describe the enhancement characteristic of effective stress; γ is the permeability damage coefficient (here γ = 0.5), and exp(γ Dt ) is the damage item to describe that the permeability becomes

3.4 Damage Tensor Calculation

37

stronger caused by damage. Through some numerical examples test including the results given below, Eq. (3.8) describing the damage effect of stress and permeability reveals excellent effect.

3.5 Wellbore Stability Analysis As the elastic properties of laminated medium, which are different in isotropic plane and transverse direction in Biot constitutive as Eq. (3.1), the strengths also vary with orientation to bedding. Weak plane model is often considered in most application cases, assuming a series of weak plane with a given orientation angle within the elastic isotropic medium, a corresponding analytical method would be established. For wellbore stability analysis, the weak plane analytical method adopts isotropic strength criterion for rock matrix and weak plane; in rock matrix the strength parameters are higher than weak plane’s, and hence it cannot serve as a purely anisotropic method. The proposed method introduces the following Pariseau strength criterion to check the strength for the transversely isotropic rock as σ˜ z2t

(σ˜ y2t z t + σ˜ z2t xt )

v

Sv2

+ T2

σ˜ z2t Cv2

+



σ˜ y2t z t + σ˜ z2t xt Sv2

−

 −

σ˜ z t (σ˜ xt + σ˜ yt ) Tv2



σz t σ˜ xt + σ˜ yt Cv2

≥ 1 tensile failure in transverse direction (3.9a) ≥ 1 compressive failure in transverse direction

(3.9b)

2 σ˜ y2t z t + σ˜ z2t xt σ˜ x2t yt − σ˜ xt σ˜ yt σ˜ xt + σ˜ yt + + ≥ 1 tensile failure in isotropic plane Sh2 Sv2 Th2

(3.9c) σ˜ y2t z t + σ˜ z2t xt σ˜ x2t yt − σ˜ xt σ˜ yt (σ˜ xt + σ˜ yt )2 + + ≥ 1 compressive failure in isotropic plane Sh2 Sv2 Ch2

(3.9d) where components of stresses with damage in Eq. (3.9) are all expressed in TCS; Th and Ch are uniaxial tensile and compressive strengths in isotropic plane respectively; Tv and Cv are uniaxial tensile and compressive strengths in transverse direction respectively; C45 is the uniaxial compressive strength measured on a sample oriented at 45° to isotropic plane; Sh and Sv are Ch Th /2(Ch + Th ) and C45 /2 respectively, and all these aforementioned strength parameters can be tested by traditional rock mechanical experiments. As shown in Fig. 3.3, the stress results of the middle section in the 3D FE model with well deviation angle and azimuth represent the stress state under the current pressure load of mud on borehole, so the Pariseau criterion is used to check the stress as Eq. (3.9). By increasing current pressure load (mud weight) and

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3 Finite Element Analysis for Continuum Damage Evolution …

repeating application of the Pariseau criterion for wellbore stability analysis, both of the collapse pressure (lower mud weight) and fracture pressure (upper mud weight) could be achieved.

3.6 Results and Discussion The proposed analysis strategy has been coded into a MATLAB program and partly uses the FE solver of COMSOL Multiphysics based on the FEA strategy. This section presents three interrelated numerical examples showing the excellent performance of the procedure. Throughout, the program is run on a DELL personal computer. The first example is chosen to discuss the effectiveness of the proposed method, by comparing the results of pore pressure and stress of rock surrounding wellbore with the porelastic analytical method, elastic drained and undrained analyses. The second example analyzes instability analysis of wellbore failure region for isotropic and transversely isotropic strength to show the significance of anisotropic analysis. In last example, a practical shale engineering case, the collapse and fracture pressure (safe mud weight) will be computed. In the three examples, the shape parameter m of Weibull distribution for heterogeneous material property and strength property are set as 5 and constant a in DP strength criterion is set as 0.29; when the FE procedure is implemented, the tetrahedral element is used and the adaptive error tolerance Tol is set as 5% throughout.

3.6.1 Pore Pressure and Stress Analysis of Rock Surrounding Wellbore In this study, a special case for laminated rock is considered with the axis of material symmetry coinciding with the axis of the wellbore model as in Fig. 3.5, in which

Fig. 3.5 Wellbore model with axis coinciding with axis of material symmetry

3.6 Results and Discussion Table 3.1 Physical parameters for example 1

39 Parameter

Value

Parameter

Value

E h / GPa

12.2

αh

0.876

E v / GPa

4.3

αv

0.946

vh

0.12

k h / nD

0.1

vv

0.08

k v / nD

0.1

φ0 /%

0.16

μ/ (Pa s)

1.0 × 10−3

analytical solutions exist (Hoang et al. 2009). The wellbore model is given inner diameter d 1 as 12.7 mm and outer diameter d 2 as 38.1 mm, and the inner boundary is unsupported and vented to the atmosphere while the outer boundary is jacketed and subjected to a uniformly confining pressure S p of 73.26 MPa as the in situ stresses. The basic physical parameters of the model are listed in Table 3.1, in which the parameter μ is pore fluid viscosity coefficient. Figure 3.6 presents pore pressure and effective stress distributions inside the model, as function of the normalized radial distance (r − d1 /2)/(d2 /2 − d1 /2), here r is the absolute radial distance from center, normalized radial distances of zero and one corresponds to the inner boundary and the outer boundary respectively.

Fig. 3.6 Results comparison of proposed method, porelastic analytical method, elastic drained and undrained analyses

40

3 Finite Element Analysis for Continuum Damage Evolution …

To further illustrate the FE solutions of the proposed HMD coupling method for this tight rock, results from the poroelastic analytical method, conventional elastic drained and undrained analysis (Hoang et al. 2009) are also included. It is evident that the elastic drained analysis gives fundamentally inaccurate results because it neglects the pore pressure generation inside rock; in general, elastic drained analysis could be oversimplified. On the other hand, the elastic undrained analysis is able to follow both the FE results of the proposed method and porelastic analytical results fairly well for the normalized radial distances in middle-field and far-field regions. However, for the near-field region which is the most prone to failure, the elastic undrained analysis fails to account for the pore fluid diffusion into the central hole and gives unacceptable results for failure analysis. Figure 3.6a shows that significant pore pressure is generated and sustained inside the laminated rock, suggesting that deformation behavior as well as failure initiation and propagation might differ significantly from other rock with higher permeability. Furthermore, Fig. 3.6b shows that the effective radial stress is tensile with significant magnitude in near-field region close to the central hole, suggesting that both spalling failure and shear collapse can be present whose interplay might be important during failure initiation as well as the subsequent yielding behavior of the rock. Figure 3.6c–d also show the good consistence of the effective tangential and axial stresses results of the proposed method with porelastic analytical and undrained elastic results.

3.6.2 Instability Analysis of Wellbore Failure Region Consider a parametric analysis with the input parameters provided in Table 3.2. The analysis is conducted by varying these two drilling direction parameters (azimuth αb and deviation angle βb ) and a given mud pressure to assess their effects on compressive and tensile failure regions. A practical wellbore stability analysis has been carried Table 3.2 Physical parameters for example 2

Parameter

Value

Parameter

Value

h/m

5500

νh

0.3

d/ mm

215.9

νv

0.3

S v / (g/cm3 )

2.26

Gv / GPa

9.14

S h / (g/cm3 )

2.13

p/ (g/cm3 )

1.08

/ (g/cm3 )

2.77

αh

0.8

αs

N45° E

αv

0.8

γt /(°)

45

k h / mD

0.03

φ t /(°)

315

k v / mD

0.003

E h / GPa

47.07

φ 0 /%

3.7

E v / GPa

15.1

μ/(Pa s)

1.84 × 10−5

ρa

0.717

SH

/ (kg/m3 )

3.6 Results and Discussion

41

out for the case of Pedernales field in Venezuela (Willson et al. 1999), whose physical parameters are shown as Table 3.2, where h and ρ a are depth and pore fluid density respectively. As shown in Table 3.2, the state of in situ stress at the depth of interest is the typical strike-slip faulting regime in which the maximum horizontal stress S H is the major principal stress and minimum horizontal stress S h is the least. In order to compare with the isotropic failure, the DP strength criterion and maximum tangential stress strength criteria are adapted as follows: a I1 +



J2 ≥ K f compressive failure

σθ ≤ −σt tensile failure

(3.10a) (3.10b)

here σθ is the effective tangential stress; σt is uniaxial tensile strength; parameters in the DP strength criterion are consistent with the introduction in Sect. 3.4. For instability analysis of compressive collapse failure region, the operational mud pressure used is 1.18 g/cm3 and the results are shown in Table 3.3. For isotropic strength, the failure regions are the same when difference between both azimuths is 180° such as 45° and 225°, 135° and 315°. However, when the anisotropic strength is incorporated into the analysis, the failures in transverse direction occur throughout as the selected four cases of βb = 90◦ , and αb = 45◦ , 225◦ , 135◦ , 315◦ , which cannot be analyzed by isotropic strength. For both isotropic and the anisotropic strength cases, it can be seen that drilling along the direction of azimuth αb = 135◦ improves wellbore stability compared to one in the direction of azimuth αb = 315◦ as the dip direction. When azimuth αb is 45° or 225°, the maximum horizontal stress SH is beneficial in terms of wellbore stability controls, and then failure zones gradually enlarge and become deeper with increasing well deviation angle βb ; however, when azimuth αb is 135° or 315°, the failure zones become narrower and shallower with increasing well deviation angle βb . On the other hand, as azimuth αb is 45° or 225°, due to wellbore drilled in the direction of minimum horizontal stress Sh having larger pressure under the normal faulting regime, it can be seen that failure regions are much deeper and more unstable than ones as azimuth αb is 135° or 315°. The aforementioned wellbore failure regions were mostly experienced drilling instability problems in Pedernales field (Willson et al. 1999). For instability analysis of tensile fracture failure region, the operational mud pressure used was 4.72 g/cm3 and the results are shown in Table 3.4. For isotropic strength, the failure regions are the same when difference between both azimuths is 180° as the compressive failure case. When the anisotropic strength is incorporated into the analysis, the failures in transverse direction occur as the selected four cases of βb = 0◦ and αb = 45◦ , 225◦ , βb = 90◦ and αb = 135◦ , 315◦ , whose failure regions are parallel to the isotropic plane; for the case of βb = 45◦ and αb = 135◦ , a special tensile failure form occurs in isotropic plane. It is worth noting that the tensile failures as ‘square borehole’ occur at four locations around the top and bottom of the wellbore in isotropic plane as the two cases of βb = 0◦ and αb = 45◦ , βb = 0◦ and

Table 3.3 Collapse failure regions around wellbore drilled with varying azimuth and deviation angle: a for case of isotropic strength, b for case of transversely isotropic strength in isotropic plane, c for case of transversely isotropic strength in transverse direction

42 3 Finite Element Analysis for Continuum Damage Evolution …

Table 3.4 Fracture failure regions around wellbore drilled with varying azimuth and deviation angle: a for case of isotropic strength, b for case of transversely isotropic strength in isotropic plane, c for case of transversely isotropic strength in transverse direction

3.6 Results and Discussion 43

44

3 Finite Element Analysis for Continuum Damage Evolution …

αb = 225◦ , which has been observed in laboratory tests and analysis (Mastin et al. 1991; Cook et al. 1994). When azimuth αb is 45° or 225°, the maximum horizontal stress SH is beneficial in terms of wellbore stability controls, and then failure zones of βb = 45◦ is narrower and shallower than the corresponding cases of βb = 0◦ and βb = 90◦ ; however, when azimuth αb is 135° or 315°, the failure zones become deeper with increasing well deviation angle βb . On the other hand, as azimuth αb is 45° or 225°, due to wellbore drilled in the direction of minimum horizontal stress Sh having larger pressure under the normal faulting regime, it can be seen that failure regions are much narrower and more stable than ones as azimuth αb is 135° or 315°.

3.6.3 Collapse and Fracture Pressure Computation This example will discuss the collapse pressure (lower critical mud weight) and fracture pressure (upper critical mud weight) computed by the proposed method using the physical parameters from practical engineering of Pedernales field in Venezuela as Example 2. This example has been analyzed by weak plane analytical method (Lee et al. 2012), in which the anisotropic rock strength characteristic is incorporated, and applied to the study of effect of anisotropic wellbore failures (Lee et al. 2013). Figures 3.7 and 3.8 show collapse and fracture pressure results with well deviation computed by the proposed method that compares with the corresponding collapse pressure results of weak plane analytical method, from which it can be seen that the results agree very well. In Fig. 3.7, the collapse pressure results with azimuth αb = 45◦ are greater than the ones with αb = 135◦ , 315◦ parallel to the dip direction, as the described reason of wellbore collapse failure above, due to wellbore drilled in the direction of maximum horizontal stress SH having larger stress differences under the normal faulting regime. Furthermore, it can be seen that drilling along the direction of azimuth αb = 315◦ Fig. 3.7 Collapse pressure results comparison of the proposed method and weak plane analytical method

3.6 Results and Discussion

45

Fig. 3.8 Fracture pressure results computed by the proposed method

needs greater mud weight (collapse pressure) for wellbore stability than the corresponding values in the directions of azimuth αb = 135◦ ; for both of azimuths αb = 135◦ , 315◦ , the in situ stresses states are the same to each other in well deviation angles of βb = 0◦ , 90◦ respectively, which leads to the same collapse pressure results computed by the proposed method or weak plane analytical method as shown in Fig. 3.7. In Fig. 3.8, the fracture pressure results computed by the proposed method with azimuth αb = 45◦ , 225◦ are greater than the ones with αb = 135◦ , 315◦ parallel to the dip direction, because the former has the greater in situ stresses state as vertical stress Sv and maximum horizontal stress SH under the normal faulting regime. As the conclusions in the aforementioned collapse pressure analysis, drilling along the direction of azimuth αb = 315◦ needs greater mud weight (fracture pressure) for wellbore stability than the corresponding values in the directions of azimuth αb = 135◦ ; for the azimuths αb = 45◦ , 225◦ and αb = 135◦ , 315◦ , the in situ stresses states are the same to each other in well deviation angles of βb = 0◦ , 90◦ respectively, which leads to the same fracture pressure results. Through the aforementioned analysis, it can be seen that the solutions computed by the proposed method show that both the wellbore failure regions and the collapse and fracture pressure have an excellent consistency.

3.7 Conclusions The FEA technology with suitable analysis strategy for effective and reliable wellbore stability analysis of transversely isotropic rock has been presented, which has successfully yielded an efficient and reliable FE procedure that obtains the results of pore pressure and stress surrounding wellbore; furthermore utilizing the technology, the wellbore failure region is analyzed and the safe mud weight for collapse and fracture pressure for transversely isotropic rock are computed. For laminated rock,

46

3 Finite Element Analysis for Continuum Damage Evolution …

comprehensive utilization of the classic Biot constitutive relation, damage tensor and Pariseau strength criterion has described the anisotropic wellbore failure behavior of being consistent with the property of rock. One of the key techniques incorporated into this procedure is the technology transfer of damage analysis for rock, isotropic continuum damage variable developed into transversely isotropic damage tensor, to form the FE model with the HMD coupling. Results for typical numerical examples, including ones known to be obvious challenge for traditionally analytical method, have shown that the present transversely isotropic analysis will obtain significantly different results from isotropic case and possess the potential for further extension to more complex mechanic problems (e.g. anisotropic wellbore stability and damage evolution problems). A recent study of anisotropic damage evolution and hydraulic fracturing model successfully simulated volume fracturing and fracture propagation in rock, thus paving the way for the corresponding anisotropic failure problems of brittle medium and practical shale extraction in complex conditions by applying the FE strategy of the proposed procedure. These advances will be reported in future publications as they are developed.

References Abousleiman YN, Hoang SK, Tran MH (2010) Mechanical characterization of small shale samples subjected to fluid exposure using the inclined direct shear testing device. Int J Rock Mech Min Sci 47(3):355–367 Abousleiman YN, Nguyen VX (2005) PoroMechanics response of inclined wellbore geometry in fractured porous media. J Eng Mech 131(11):1170–1183 Bažant ZP, Salviato M, Chau VT, Visnawathan H, Zubelewicz A (2014) Why fracking works. J Appl Mech 81(10):1–10 Benjamin L, Jean H (1986) Accurate numerical solutions for drucker-prager elastic-plastic models. Comput Methods Appl Mech Eng 54(3):259–277 Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164 Biot MA (1954) Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J Appl Phys 25(11):1385–1391 Chen SL, Abousleiman YN (2016) Stress analysis of borehole subjected to fluid injection in transversely isotropic poroelastic medium. Mech Res Commun 73(4):63–75 Cheng AHD (1997) Material coefficients of anisotropic poroelasticity. Inter J Rock Mech Min Sci 34(2):199–205 COMSOL Inc (2010) COMSOL Multiphysics user’s guide Cook JM, Goldsmith G, Bailey L, Audibert A, Bieber MT (1994) X-ray tomographic study of the influence of bedding plane orientation on shale swelling. SPE annual technical conference and exhibition. Soc Pet Eng SPE-28061-MS Detourmay E, Cheng AHD (1993) Fundamentals of poroelasticity. Pergamon, Oxford Hill R (1950) The mathematical theory of plasticity. Oxford University Press, Oxford Hoang S, Abousleiman YN, Ewy RT (2009) Openhole stability and solids production simulation in emerging reservoir shale using transversely isotropic thick wall cylinders. SPE annual technical conference and exhibition. Soc Pet Eng SPE-135865-MS Jaeger JC, Cook NGW, Zimmerman R (2009) Fundamentals of rock mechanics. Wiley, Neywork Jean L (1985) Coupled elasto-plasticity and damage constitutive equations. Comput Methods Appl Mech Eng 51(1):31–49

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Krajcinovic D (2000) Damage mechanics: accomplishments, trends and needs. Inter J Solids Struct 37:267–277 Krajcinovic D, Lemaitre J (1987) Continuum damage mechanics theory and applications. USA, Springer, New York Lee H, Chang C, Ong SH, Song I (2013) Effect of anisotropic borehole wall failures when estimating in situ stresses: a case study in the Nankai accretionary wedge. Mar Pet Geol 48:411–422 Lee H, Ong SH, Azeemuddin M, Goodman H (2012) A wellbore stability model for formations with anisotropic rock strengths. J Pet Sci Eng 96:109–119 Liang ZZ, Tang CA, Li HX, Xu T, Yang TH (2005) A numerical study on failure process of transversely isotropic rock subjected to uniaxial compression. Rock Soil Mech 26(1):57–62 ((In Chinese)) Liao JH, Zhuang Z (2012) A consistent projection-based SUPG/PSPG XFEM for incompressible two-phase flows. Acta Mech Sinica 28(5):1309–1322 Lin ZJ, Zhuang Z (2014) Enriched goal-oriented error estimation for fracture problems solved by continuum-based shell extended finite element method. Appl Math Mech 35:33–48 Lo T, Coyner KB, Toksöz MN (1986) Experimental determination of elastic anisotropy of berea sandstone, chicopee shale, and chelmsford granite. Geophysica 51(1):164–171 Lu YH, Chen M, Jin Y, Zhang GQ (2012) A mechanical model of borehole stability for weak plane formation under porous floe. Pet Sci Tech 30:1629–1638 Lu YL, Elsworth D, Wang LG (2013) Microcrack-based coupled damage and flow modeling of fracturing evolution in permeable brittle rocks. Comput Geotech 49:226–244 Mastin LG, Heinemann B, Krammer A, Fuchs K, Zoback MD (1991) Stress orientation in the KTB pilot hole determined from wellbore breakouts. Sci Drill 2:1–12 Nguyen VX, Abousleiman YN (2009) Naturally fractured reservoir three-dimensional analytical modeling: theory and case study. SPE annual technical conference and exhibition. Soc Pet Eng SPE-123900-MS Pariseau WG (1968) Plasticity theory for anisotropic rocks and soils. Proceedings of tenth symposium on rock mech. pp 267–295 Roberto SR, Chaitanya D, Yi KY (2009) Unlocking the unconventional oil and gas reservoirs: the effect of laminated heterogeneity in wellbore stability and completion of tight gas shale reservoirs. Offshore tech conference Wang T, Gao Y, Liu ZL, Wang YH, Yang LF, Zhuang Z (2014) Numerical simulations of hydraulic fracturing in large objects using an extended finite element method. J Tsinghua Uni (Sci Tech) 10:1304–1309 Wang YL, Liu ZL, Lin SC, Zhuang Z (2016) Finite element analysis of seepage in rock based on continuum damage evolution. Eng Mech 33(11):29–37 Warpinski NR, Mayerhofer MJ, Vincent MC, Vincent M, Mayerhofer M (2009) Stimulating unconventional reservoirs: maximizing network growth while optimizing fracture conductivity. J Can Pet Tech 48(10):39–51 Willson SM, Last NC, Zoback MD, Moos D (1999) Drilling in South America: a stability approach for complex geological conditions. Latin American and caribbean petroleum engineering conference. Soc Pet Eng SPE-53940-MS Wu Y, Liu J, Elsworth D, Chen ZW, Connell B, Pan ZJ (2010) Dual poroelastic response of a coal seam to CO2 injection. Inter J Greenhouse Gas Control 4(4):668–678 Xiao Y, Liu H, Desai CS, Sun YF, Liu H (2016) Effect of intermediate principal stress ratio on particle breakage of rockfill material. J Geotech Geoenviron Eng 142:06015017 Xiao Y, Sun Y, Hanif KF (2015) A particle-breakage critical state model for rockfill material. Sci China Tech Sci 58(7):1125–1136 Xu DD, Liu ZL, Liu XM, Zeng QL, Zhuang Z (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Comput Mech 54(2):489–502 Yang HL, Shen RC, Fu L (2013) Composition and mechanical properties of gas shale. Pet Drill Tech 41(5):31–35 ((In Chinese))

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Zeng QL, Liu ZL, Xu DD, Zhuang Z (2014) Modeling stationary and moving cracks in shells by X-FEM with CB shell elements. Sci China Tech Sci 57(7):1276–1284 Zhang H, Liu J, Elsworth D (2008) How sorption-induced matrix deformation affects gas flow in coal seams: a new FE model. Inter J Rock Mech Min Sci 45(8):1226–1236 Zhu WC, Bruhns OT (2008) Simulating excavation damaged zone around a circular opening under hydromechanical conditions. Inter J Rock Mech Min Sci 45(5):815–830 Zhuang Z, Cheng BB (2011) A novel enriched CB shell element method for simulating arbitrary crack growth in pipes. Sci China Phys Mech Astron 54(8):1520–1531 Zhuang Z, Liu ZL, Wang YL (2015) Fundamental theory and key mechanical problems of shale oil gas effective extraction. Chin Q Mech 33(1):8–17 ((In Chinese))

Chapter 4

Finite Element Analysis for Continuum Damage Evolution and Inclined Wellbore Stability of Transversely Isotropic Rock Considering Hydro-Mechanical-Chemical Coupling

4.1 Introduction The wellbore instability problems appeare in shale gas exploitation due to the anisotropic mechanical behaviors of the unconventional shale (Zhuang et al. 2015). Shale exhibits typical water-sensitive and laminated characteristics. By experiment researches, we know the shale is composed of fabric-related and plate-like minerals (Lo et al. 1986; Abousleiman et al. 2010), in which material and strength parameters possess apparent heterogeneity and typical transversely isotropic behavior (Yang et al. 2013b). Chemically active effect of drilling fluid is considered as the important influencing factor for wellbore stability of mud shale during drilling (Nguyen and Abousleiman 2009). The stability problem of a wellbore drilled into a thinly laminated anisotropic rock formation in this chapter is shown in Fig. 4.1, which is assumed in transversely isotropic formation coordinate system (TCS) xt − yt −z t ; the plane parallel and direction vertical to bedding are frequently replaced below by the terminologies as isotropic plan and transverse direction respectively. Drilling a borehole with a given mud pressure P into formation, which is fully saturated with a pore fluid and subjected to the preexisting in situ stresses (SH is the maximum horizontal principal stress; Sh is the minimum horizontal principal stress; Sv is the vertical stress) initially in static equilibrium, disturbs the state of stress in the vicinity of the borehole. The finite element analysis (FEA) technology presented in this chapter takes watersensitive and laminated rock as the research objects (a typical porous medium with transversely isotropic property and strength, dispersedly distributed damage region under applied mud pressure and in situ stresses), to obtain the collapse cycling time, collapse and fracture pressure as safe mud weight. Shale has the characteristic of low porosity and permeability as unconventional rock; particularly the key technology of horizontal well drilling can create damage and fracture propagation around the wellbore, to change the porosity and permeability (Bažant et al. 2014). The effective stress of porous medium will change with the fluid flow, pressure diffusion in pores and solid deformation correspondingly. In other words, hydro-mechanical (HM) coupling makes the response of the rock © Science Press 2021 Y. Wang, Adaptive Analysis of Damage and Fracture in Rock with Multiphysical Fields Coupling, https://doi.org/10.1007/978-981-15-7197-8_4

49

50

4 Finite Element Analysis for Continuum Damage Evolution …

Fig. 4.1 Wellbore in water-sensitive and laminated rock formation subjected to in situ stresses

reflect complex time-dependent effect obviously. For some conventional and simple problems, such as possessing regular solving domain with homogeneous material property, analytical method via the poroelastic theory has been proposed to solve a number of multiphysical fields coupling, wellbore stability and fracture problems (Abousleiman and Nguyen 2005; Nguyen and Abousleiman 2009), and a related method for an inclined borehole in a special transversely isotropic formation has been developed (Hoang et al. 2009; Chen and Abousleiman 2016). Utilizing the poroelastic analytical solutions and introducing the isotropic rock strength criterion, a weak plane model for wellbore stability of anisotropic formations had been developed (Lu et al. 2012). Rock is a typically heterogeneous, brittle medium with complex damage and breakage occurring under external load (Xiao et al. 2016; Zhao et al. 2016); however, the analytical method unfortunately does not consider heterogeneity, plastic deformation and damage evolution, and hence cannot serve as a practical and general method. Therefore, some researchers were motivated to probe into the numerical area of anisotropic rock fracture, damage analysis and wellbore stability problems. Some researchers have made a series of achievements on the basic theory and algorithm of extended finite element method (XFEM) as discrete fracture network model for arbitrary crack growth (Zhuang and Cheng 2011; Zeng et al. 2014). On the other hand, some researchers proposed the continuous model based on the damage mechanics to analyze the failure process. Compared with discrete fracture network model, the continuous model more easily handles the multiphysical fields coupling and complex medium containing pores and fractures problems to be consistent with the material behavior (Krajcinovic and Lemaitre 1987). Inspired from which, some numerical methods and simulation technologies based on damage analysis were proposed (Zhu and Bruhns 2008; Lu et al. 2013). The damage variable discribing the compressive and tensile behavior of rock was developed by mesoscopic damage mechanics (Zhu and Bruhns 2008). A novel and simple FEA technology of seepage in rock has been presented by the author of this book with the further development of the continuous model and continuum damage variable (Wang et al. 2016a, b). The continuum damage variable was developed effectively into a damage tensor introducing the stresses state under the current load and anisotropic strength to consider the bearing capacity; utilizing the damage tensor, this method developed finite element (FE) algorithm to obtain the stress solutions with damage,

4.1 Introduction

51

and was applied to the wellbore stability analysis of transversely isotropic rock with hydro-mechanical-damage (HMD) coupling (Wang et al. 2017a). In resent research work, the author of this book developed the FEA technology of transversely isotropic rock with hydro-mechanical-chemical-damage (HMCD) coupling for simulating the behavior of anisotropic damage evolution (Wang et al. 2017b). Many researchers dedicated to the anisotropic strength criteria (Mortara 2010; Xiao et al. 2012). Hill (1950) proposed a general criterion expressed as a quadratic function for anisotropic materials, which was not directly applicable to geological materials because the strength behavior of most geological materials was dependent on the hydrostatic stress. Pariseau (1968) extended Hill criterion to account for the effect of the hydrostatic stresses. For wellbore stability, weak plane strength criterion (Lee et al. 2012, 2013) was considered as a simple, practical and effective criterion. Wellbore stability problems of laminated rock were analyzed using FEM based on the weak plane strength criterion to analyze the failure in rock matrix and weak plane, which was introduced in this chapter to check the failure. The FEA technology with HMCD coupling of rock is proposed for water-sensitive and laminated rock in this chapter. In the analysis strategy, a numerical threedimensional (3D) FE model with in situ stress boundary conditions (BCs) for declined borehole with different inclination and azimuth angle was developed, and FE solutions were obtained using Biot constitutive theory (Biot 1941, 1954) and considering hydration effect. Then the effective stress with damage was utilized to obtain both of the collapse pressure (lower mud weight) and fracture pressure (upper mud weight) based on weak plane strength criterion. This yields an FEA technology that is able to establish transversely isotropic rock model and analyze wellbore stability in petroleum engineering applications. Some numerical examples including the below examples have shown that the proposed FEA technology is effective and feasible for inclined wellbore stability of water-sensitive and laminated rock.

4.2 Transverse Isotropy and Hydration Characterization 4.2.1 Constitutive Equation The rock reservoir has the typical nonlinear characteristics plastic deformation and damage evolution; however, the nonlinear model will increase the difficulty and calculation cost for analysis, so the linear elastic models can be used in most of the problems. The proposed method in this chapter takes the hypothesis of linear elasticity and small deformation, and introduces the damage analysis in the study to describe the nonlinear behavior below. In another hand, the Biot constitutive relation is a well-established constitutive relation for elastic pore medium (Detourmay and Cheng 1993), which in the proposed method will derivate a coupled HM model. The anisotropic Biot constitutive equation (Cheng 1997) can be expressed as σ¯ e = σ − α p = C¯ε − α p

(4.1)

52

4 Finite Element Analysis for Continuum Damage Evolution …

where p is the pore pressure; σ¯ e = σe +σe is the effective stress vector; ε¯ = ε+ε is T  the strain vector; σe and ε are effective stress vector σxet σ yet σzet τxet yt τ yet zt τxet zt and  T strain vector εxt ε yt εzt γxt yt γ yt zt γxt zt is respectively without considering hydration characterization; σe and ε are effective stress and strain vector respectively introduced by hydration; the compressive stress and strain are positive throughout in this chapter and vice versa; α is Biot coefficient vector { αx α y αz 0 0 0 } T ; in transversely isotropic case, αx and α y are equal parameters in isotropic plane expressed as αh and αz is parameter in transverse direction expressed as αv in below content; p is the pore pressure; C is the stiffness matrix. For general anisotropy, the constitutive relation contains twenty-eight independent material coefficients. For materials with three mutually orthogonal planes of elastic symmetry, known as orthotropic, there exist thirteen independent material coefficients; furthermore, stiffness matrix C of transversely isotropic case can be simplified as ⎡

M11 M12 M13 ⎢ M11 M13 ⎢ ⎢ M33 ⎢ C=⎢ ⎢ ⎢ ⎣ sym.

0 0 0 M44

0 0 0 0 M55

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ M55

where drained elastic modulus can be expressed in terms of engineering constants as M11 =

E h (E v − E h νv2 ) (1 + νh )(E v − E v νh − 2E h νv2 )

E h (E v νh − E h νv2 ) (1 + νh )(E v − E v νh − 2E h νv2 ) E h E v νv = E v − E v νh − 2E h νv2

M12 = M13

E v2 (1 − νh ) E v − E v νh − 2E h νv2 Eh = 2(1 + νh )

M33 = M44

M55 = G v where E h and νh are drained Young’s modulus and Poisson’s ratio in the isotropic plane; E v and νv are similar quantities related to the direction of the axis of symmetry (transverse direction); and G v is the shear modulus related to the direction of the axis of symmetry. For the convenience of introducing below, the elastic and permeability matrixes can be expressed as

4.2 Transverse Isotropy and Hydration Characterization

53

⎤ ⎤ ⎡ E xt G xt yt G xt zt k xt k xt yt k xt zt ⎥ ⎥ ⎢ ⎢ e=⎣ E yt G yt zt ⎦, kt = ⎣ k yt k yt zt ⎦ sym. E zt sym. kzt ⎡

where E xt and E yt have the equally initial value E h ; E zt has the initial value E v ; G xt yt has the initial value G h = E h /2(1 + νh ); G xt zt and G yt zt have the initial permeability G v ; k xt and k yt have the equally initial permeability kh in isotropic plane; k zt has the initial permeability kv in transverse direction; as the non-diagonal parameters, k xt yt and k xt zt have the initial permeability kh ; k yt zt has the initial permeability kv .

4.2.2 Hydration Effect Considering the chemically active effect of rock, some hydration analyses have been carried out for isotropic rock media. The chemical activity of the drilling fluid is considered as phenomenological hydration behavior, which will be introduced in this chapter. According to the conservation of fluid mass, the water absorption diffusion equation can be established, and with the boundary conditions, the moisture content and the relationship with other physical parameters at any position and time in rock can be obtained (Huang et al. 1995; Dokhani et al. 2015). The proposed method in this chapter will extend the hydration analysis to the transversely isotropic media form isotropic case; furthermore, the moisture content in isotropic plane and transverse direction can be expressed as wk (r, t¯) = w∞ + (wd/2 − w∞ )er f c(r/2 ck t¯)

(4.2)

where w k (r, t¯) is moisture content; r is the absolute radial distance from wellbore wall; t¯ is the time of water absorption diffusion; wd/2 and w∞ are the initial moisture content at wellbore wall and infinity respectively; ck is the adsorption diffusion constant (ch and cv for isotropic plane and transversely isotropic, respectively); er f c( ) is the error compensation function. On the current moisture content, the hydration expansion strain, elastic and strength parameters with moisture content of isotropic rock media can be established, in which using experimental data to form empirical formulas is the effective and practical method in the present researches (Huang et al. 1995). The proposed method will apply this technology to the transversely isotropic media, the hydration expansion strain, elastic, Poisson’s radio and strength parameters can be expressed as εi j (r, t¯) = δi j [cε1 wk (r, t¯) + cε2 wk2 (r, t¯)]

(4.3a)

ei j (r, t¯) = ei j0 ex p{ − ce [wk (r, t¯) − w∞ ]0.5 }

(4.3b)

54

4 Finite Element Analysis for Continuum Damage Evolution …

νi (r, t¯) = νi0 + cν wk (r, t¯)

(4.3c)

ηk (r, t¯) = ηk0 − cη [wk (r, t¯) − w∞ ]

(4.3d)

where εi j (r, t¯) is the coefficient in hydration expansion strain matrix ε; cε1 and cε2 are hydration strain constants; δi j denotes the Kronecker delt function; ei j is the coefficient in elastic matrix e, and ei j0 is its initial value; ce is hydration elastic coefficient; νi (r, t¯) is hydration Poisson’s radio, and νi0 is its initial value; cν is hydration constant for Poisson’s radio; ηk (r, t¯) is hydration strength; ηk0 is initial value; cη is hydration strength constant. The above i and j take 1, 2 and 3; each hydration constant can be determined according to experiments. In order to investigate the chemical characteristic for moisture content w k (r, t¯), Young’s modulus ei j (r, t¯) and strength ηk (r, t¯) with time t¯, here the dimensionless values in Eq. (4.3) adapting ck = ce = cη = w∞ = ei j0 = ηk0 = 1 and wd/2 = 2, the hydration function curves can be obtained as shown in Fig. 4.2. It can be seen that

Fig. 4.2 Hydration function curves

4.2 Transverse Isotropy and Hydration Characterization

55

the moisture content w k (r, t¯) is becoming bigger; elastic modulus ei j (r, t¯) and strength parameter ηk (r, t¯) are becoming smaller as time increases. At the short hydration time t¯ = 0.1, the variables near the wellbore wall significantly change obviously, and the surrounding rock appears change as a result of long-term effect. So the chemically active effect of water-sensitive rock needs to be considered, and the time-dependent variables should be solved effectively and reliably.

4.3 Finite Element Analysis Strategy The FEA technology in this chapter achieves the results for wellbore stability analysis of transversely isotropic rock with HMCD coupling by implementing the three-step FEA strategy as below, and the corresponding algorithm is illustrated in Fig. 4.3. (1) FE solution. For the wellbore stability problems of laminated rock, the numerical 3D FE model will be established with in situ stress boundary conditions (BCs) for declined borehole with different inclination and azimuth angle, in which the rock media considers the hydration effect. Under the current pressure load Pn of mud on borehole, the effective stress σet and permeability kt can

Fig. 4.3 FEA algorithm flowchart of wellbore stability analysis for collapse and fracture pressure computation at time t¯

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4 Finite Element Analysis for Continuum Damage Evolution …

be calculated to obtain solutions at time t¯ by the standard FEA technology, as described in Sect. 4.4. (2) Damage tensor calculation. The aforementioned effective stress σet and the compressive or tensile strength are used to calculate the damage tensor Dt ; furthermore the stress σ˜ t and permeability kt with damage can be calculated respectively, by which the damage state of the rock under the current hydration and load can be described in Sect. 4.5. (3) Wellbore stability analysis. Utilizing the weak pane strength criterion, the stress σ˜ t with damage will be used to check whether the compressive or tensile situation happens in the weak plane and rock matrix of laminated rock as described in Sect. 4.6. The pressure load Pn of mud on borehole will increase a load step P and return to the first step (i.e., FE solution) until both of the collapse pressure (lower mud weight) load interval (Pcl , Pcu ) and fracture pressure (upper mud weight) load interval (Pfl , Pfu ) at time t¯ are all achieved. Because the load intervals (Pl , Pu ) for collapse pressure as (Pcl , Pcu ) or fracture pressure as (Pfl , Pfu ) can be too big to obtain the reliable results, the dichotomy method was adapted for selecting the intermediate load Pa = (Pl + Pu )/2 to update the upper or lower bound to narrow the intervals, until difference between the upper bound and lower bound is less than the user-preset error tolerance T ol, i.e., Pu − Pl ≤ T ol. The aforementioned three steps constitute a round of FE solving, damage tensor calculation and wellbore stability analysis, by which the instability mode, collapse and fracture pressure of transversely isotropic rock can be obtained to exhibit that there would be obviously different characteristics with isotropic case. The numerical examples below show that the proposed FEA strategy is effective, reliable and effective.

4.4 Finite Element Solution 4.4.1 Finite Element Formulation Utilizing the aforementioned Biot constitutive equation and the hydration analysis technology, the solid and seepage control equations can be discretized utilizing FEM as Ku = P (Ht¯ + S)pt¯

¯

i−1 +t

(4.4a)

= Spt¯

i−1

+F

(4.4b)

where K is the time-dependent stiffness matrix for hydration effect; u is the displacement vector on the FE nodes and P is the load vector; H is conduction matrix; S

4.4 Finite Element Solution

57

is memory matrix; F is the fluid convergence vector; p is the pore pressure; here t¯ expressed as superscript is time; t¯ is the time step; pt¯ is the pore pressure at i−1 t¯i−1 ; pt¯ +t¯ is the pore pressure at t¯i−1 + t¯. The load P in Eq. (4.4a) considers i−1 the pore pressure p as coupling relationship, and the pore pressure p is iterative FE solution in Eq. (4.4b). In the proposed method, the transient FE solver of COMSOL Multiphysics (COMSOL Inc. 2010) is taken to obtain the solutions at time t¯.

4.4.2 Finite Element Model and Adaptive Mesh Refinement for Wellbore Stability Analysis The inclined wellbore stability model involves multiple stress transformations between the different reference coordinate systems, further four reference coordinated systems are defined as described in Appendix: global coordinate system (GCS), in situ stress coordinate system (ICS), borehole coordinate system (BCS), and transversely isotropic formation coordinate system (TCS, as mentioned in Fig. 3.2). Each of these and their mutual relationships are shown in Fig. 4.12. Based on the coordinate systems and their relationships defined, the rotation matrices are obtained which are needed to transform stress components from one reference coordinate system to the other (Wang et al. 2017a). The 3D FE model with boundary conditions (BCs) is introduced as shown in Fig. 4.4, which is used for analysis of transversely isotropic rock considering hydromechanical coupling in Chap. 3. Furthermore, the adaptive mesh refinement for FE model of inclined wellbore is also introduced as shown in Fig. 4.5, and the FE model considers the hydration effect, and specified error tolerance Tol is set as 5%. The well-developed Weibull distribution technique is used to characterize the heterogeneous material properties (i.e., Young’s modulus E h and E v , shear modulus Gv , permeability k h and k v ) and strength properties of rock as follows (Wang et al. 2016a, 2017a, b):

Fig. 4.4 FE model and stability analysis plane of inclined wellbore with hydration effect

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4 Finite Element Analysis for Continuum Damage Evolution …

Fig. 4.5 Adaptive mesh refinement for FE model of inclined wellbore

 m−1 f (x, η, m) =

m η

x η

0,

  m  , x ≥0 exp − ηx x 3, and wh ∈ C 1 denotes the conventional FE solution on the given mesh π , in which C 1 is the space of functions that are continuous up to their first-order derivative. As in common practice, the shape functions for wh are Hermite polynomials. Given an arbitrary trial value ωa as the shift value, Eq. (5.6) can be equivalently written in the shifted form (Zienkiewicz and Taylor 2000):

5.3 Adaptive Analysis

81

Ka D = μ M D with Ka = K − ωa2 M, μ = ω2 − ωa2

(5.7)

In the proposed method, the convectional FE computation for eigenpair solutions is based on the Sturm sequence property (Clough and Penzien 1993), which can be expressed as K − ω2 M = LD(ω)LT

(5.8)

where L is a lower triangular matrix with leading diagonal elements being one; LT is its transpose, and D(ω) is a diagonal matrix in which the number of eigenvalues less than the arbitrary trial value ωa equals the number of negative leading diagonal elements in D(ωa ). The Rayleigh quotient is used to accelerate the convergence on the eigenvalues: ω2 =

DT K D DT M D

(5.9)

Utilizing the above Sturm sequence property and the convectional bisection method (Clough and Penzien 1993), the intervals of each eigenvalue can be determined, and the inverse iteration technique is successfully introduced to compute the eigenpairs (Yuan et al. 2013, 2017). Based on these considerations, the following inverse iteration procedure is adopted: ⎧ ¯ i+1 = Ka−1 M Di D ⎪ ⎪ ⎪ ⎪ ⎪ ¯ T M Di ⎪ ⎨μ = D i+1 i+1 ¯T MD ¯ i+1 D i+1 ⎪ ⎪ ⎪ ¯ i+1 ⎪ D ⎪ ⎪ ⎩ Di+1 = sgn(μi+1 ) ¯ i+1 ) max(D

(5.10)

where i is the loop index. The above inverse iteration procedure is terminated when the following conditions are met:     μ − μ  < T ol and max D  < T ol i+1 i i+1

(5.11)

After the above inverse iteration converges, an FE solution (μh , Dh ) (i.e., (ωh , Dh ) where (ωh )2 = ωa2 + μh ) is obtained. However, the current mesh may not be sufficiently fine, in which case the accuracy of this FE solution must be estimated by a more accurate solution, namely, the superconvergent solution, which is discussed in the following section.

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5 Adaptive Finite Element Algorithm for Damage …

5.3.2 Error Estimation and Mesh Refinement The superconvergent patch recovery displacement technique was developed for computation of superconvergent displacements for FE solutions of static and dynamic problems (Wiberg et al. 1999a, b). The displacements provided by the superconvergent computation technique can be applied to eigenfunctions. For example, as shown in Fig. 5.6, element e is the superconvergent computation element, and elements e − 1 and e + 1 are neighbouring elements, in which FE nodes j − 1, j, j + 1, and j + 2 are selected for computation. The superconvergent displacements for element e can be computed as w∗ (x) =

r 

Ni (x)wih +

i=1

s 

Ni (x)w¯ i∗

(5.12)

i=1

where r(= 2) is the number of end nodes; s is the number of internal nodes; Ni (x) is the shape function. The degree of the shape function is improved by one order as r + s = m + 1. To make the best use of the superconvergent order O(h 2m ) for displacements at end nodes, the displacement recovery field can be expressed by FE nodes as: w¯ ∗ (x) = Pa

(5.13)

where P is the given function vector, and a can be obtained by the least squares fitting technique for the coincidence of displacements at the end nodes in the recovery field and the conventional FE field. The superconvergent displacements at the end nodes in recovery field w¯ ∗ (x) are used in Eq. (5.12) to obtain the superconvergent solutions on element e. Because the accuracy of the superconvergent solution w∗ is at least one order higher than that of wh , for elements of degree m > 3, a very simple strategy for error estimation is to use w∗ instead of the exact solution w to estimate the errors in wh . This error estimation method has shown good reliability and effectiveness (Wiberg et al. 1999a, b). The superconvergent solutions of Eq. (5.13) can be used in the Rayleigh quotient (Clough and Penzien 1993) to obtain estimates of the eigenvalue: Fig. 5.6 Computation of superconvergent displacements for element e

5.3 Adaptive Analysis

83

 ωk∗

=

a(wk∗ , wk∗ ) , k = 1, 2, . . . , 2n b(wk∗ , wk∗ )

(5.14)

l l where a(w, v) = 0 E I (x)w v d x and b(w, v) = 0 m(x)wv d x are the strain energy inner product and the kinematic energy inner product, respectively. The estimated eigenvalue is a stationary value when taking over all possible functions that satisfy the essential BCs. The stationary values computed by Eq. (5.14) are superconvergent eigenvalues and the corresponding functions ωk∗ are the superconvergent solutions. The Rayleigh quotient, Eq. (5.14), can be expressed based on elements as

 b ∗ ∗  e  e a E I (x)wk wk d x ∗ , k = 1, 2, . . . 2n ωk = e b e e ae m(x)wk wk d x

(5.15)

where ae and be are the end nodes of the boundary for element e. To determine if the solution on the current mesh satisfies the given tolerance, the error is estimated by      1/ 2  ∗  e  ≤ T ol · wh 2 + e∗ 2 ne , k = 1, 2, . . . , 2n w,k e k w,k

(5.16)

∗ where ew,k is the error of the superconvergent displacements wk∗ and the computed displacements wkh ; n e is the number of elements; w = [a(w, w)] 1/2 . Equation (5.16) can be equivalently written in the following form:

ξk =

 *  e 

w,k e

e¯w,k

with e¯w,k = T ol ·

 2    1 2 wh  + e* 2 n / , k w,k e

k = 1, 2, . . . , 2n

(5.17)

where ξk should satisfy: ξk ≤ 1, k = 1, 2, . . . , 2n

(5.18)

Usually it is more than sufficient to set M in the range of 4 ≤ M ≤ 8. Therefore, without loss of generality, M is set to 6 for the remainder of this chapter. If Eq. (5.18) is not satisfied for any interior point, the corresponding element needs to be subdivided into uniform sub-elements by inserting some interior nodes through h-refinement (Zienkiewicz and Zhu 1992a, b; Zienkiewicz and Taylor 2000), which are calculated by −1 m h k,new = ξk / h k,old , k = 1, 2, . . . , 2n

(5.19)

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5 Adaptive Finite Element Algorithm for Damage …

 Fig. 5.7 Uniform subdivision of element e (e.g., h k,new = h k,old 6)

where h k,new is the length of the sub-element; and h k,old is the original length of element e. The above element subdivision approach is implemented as    1 m , d , k = 1, 2, . . . , 2n n k,new = min ξk /

(5.20)

where n k,new is the number of subelements after element subdivision; and d is the limit number for avoiding too many redundant elements. Each element e that does not satisfy the pre-specified error tolerance is uniformly subdivided, e.g., h k,new =  h k,old 6 as shown in Fig. 5.7.

5.4 Newton-Raphson Iteration Based on the frequency measurement method, the residuals of the frequencies and cracks are consistent with each other; therefore, the Newton-Raphson iteration technique can be introduced (Clough and Penzien 1993). For the identification of n cracks in a beam, there should be 2n unknown crack parameters: α1 , β1 , α2 , β2 , …, αn , and βn . To match the number of equations and the number of unknown parameters, it 0 0 and ω2n are is assumed that 2n natural frequency measurements ω10 , ω20 , …, ω2n−1 available in advance. The Newton-Raphson iteration procedure is applied in this chapter as follows: (1) The user assumes initial values of α1 , β1 , α2 , β2 , …, αn , and βn and the FE mesh of the beam. (2) Locate the positions that represent the cracks according to the new crack locations parameters β1 , β2 , …, βn . h with (3) Solve the forward eigenproblem to obtain FE solutions for ω1h , ω2h ,…,ω2n the crack parameters α1 , β1 , α2 , β2 , …, αn , and βn , and evaluate the Jacobian matrix:

5.4 Newton-Raphson Iteration

⎡

85 ∂ω1h ∂α1 ∂ω2h ∂α1

∂ω1h ∂β2 ∂ω2h ∂β2

···

h h h h ∂ω2n ∂ω2n ∂ω2n ∂ω2n ∂α1 ∂β1 ∂α2 ∂β2

···

⎢ ⎢ ⎢ J=⎢ . ⎢ . ⎣ .

∂ω1h ∂β1 ∂ω2h ∂β1

.. .

∂ω1h ∂α2 ∂ω2h ∂α2

.. .

.. .

···

∂ω1h ∂αn ∂ω2h ∂αn

.. .

∂ω1h ∂βn ∂ω2h ∂βn

⎤

⎥ ⎥ ⎥ .. ⎥ ⎥ . ⎦

(5.21)

h h ∂ω2n ∂ω2n ∂αn ∂βn

and compute the residuals of frequencies: Rk = ωkh − ωk0 , k = 1, 2, . . . , 2n,

(5.22)

(4) Solve the following equation by Newton–Raphson iteration:

J dCT = −RT

(5.23)

where CT = (α1 , β1 , α2 , β2 , . . . , αn , βn )T and RT = (R1 , R2 , . . . , Rn )T , through which the residuals of n cracks (dαi , dβi ) (i = 1, 2, . . . , n) will be obtained. (5) Update the crack parameters by utilizing the residuals of cracks:

(αi )new = (αi )old + dαi , (βi )new = (βi )old + dβi , i = 1, 2, . . . , n

(5.24)

where ( )new and ( )old represent the new and old cracks in the last step, respectively. Update each old crack with the new one, as shown in Fig. 5.8. (6) In the new crack condition, return to step (1) and repeat the loop until the residuals of frequencies become sufficiently small. Note that the FE mesh of conventional FEM for the Newton-Raphson iteration procedure is determinate without mesh refinement. However, the adaptive FE analysis proposed in this chapter will have more accurate results and a better convergence rate compared to the conventional FE analysis, which is shown in the numerical examples in Sect. 5.7. Fig. 5.8 Update of size and location of cracks in one iteration step

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5 Adaptive Finite Element Algorithm for Damage …

5.5 Damage Refinement The matrix elements in the Jacobian matrix J are related to the frequencies and crack parameters because the most widely-used method, damage detection based on the optimization theory, is reduced to a linearized system of equations. The matrix elements of the Jacobian matrix J are the sensitivities of the natural frequencies with respect to the crack parameters. Morassi (2001) developed an explicit expression of the frequency sensitivity to damage, assuming that the sizes of the cracks were sufficiently small. In this study, however, the cracks are not assumed to be small, and the elements of the Jacobian matrix J are computed numerically introducing the method by Lee (2009). For example, ∂ω1 /∂α1 and ∂ω1 /∂β1 are computed, respectively, as follows: ∂ω1 ω (α + δ, β1 , α2 , . . . , βn ) − ω1 (α1 , β1 , α2 , . . . , βn ) , |δ|