Porous Rock Fracture Mechanics. with Application to Hydraulic Fracturing, Drilling and Structural Engineering [1st Edition] 9780081007822, 9780081007815

Table of contents :
Content:
Front-matter,Copyright,Dedication,List of contributors,Preface,IntroductionEntitled to full textPart I: Introduction1 - Application of rock failure simulation in design optimization of the hydraulic fracturing, Pages 3-23
2 - Anisotropic poroplasticity in saturated porous media, effect of confining pressure, and elevated temperature, Pages 27-46
3 - Coupling in hydraulic fracturing simulation, Pages 47-62
4 - Stress-induced permeability evolutions and erosion damage of porous rocks, Pages 63-92
5 - Hydraulic fracture growth in naturally fractured rock, Pages 93-116
6 - Cohesive zone models, Pages 119-144
7 - Application of discrete element approach in fractured rock masses*, Pages 145-176
8 - The embedded finite element method (E-FEM) for multicracking of quasi-brittle materials, Pages 177-196
9 - Application of continuum damage mechanics in hydraulic fracturing simulations, Pages 197-212
10 - Multiscale modeling approaches and micromechanics of porous rocks, Pages 215-232
11 - Dynamic fracture mechanics in rocks with application to drilling and perforation, Pages 233-256
12 - Stability, accuracy, and efficiency of numerical methods for coupled fluid flow in porous rocks, Pages 257-283
13 - True triaxial failure stress and failure plane of two porous sandstones subjected to two distinct loading paths, Pages 285-307
Index, Pages 309-317

Citation preview

Porous Rock Fracture Mechanics

Related titles Engineering Rock Mechanics: An Introduction to the Principles (ISBN: 978-0-08-043864-1) Petroleum Related Rock Mechanics: 2nd Edn (Developments in Petroleum Science) (ISBN: 978-0-444-50260-5) Structural Geology: The Mechanics of Deforming Metamorphic Rocks (ISBN: 978-0-12-407820-8)

Woodhead Publishing Series in Civil and Structural Engineering

Porous Rock Fracture Mechanics with Application to Hydraulic Fracturing, Drilling and Structural Engineering

Edited by

Amir K. Shojaei Jianfu Shao

Woodhead Publishing is an imprint of Elsevier The Officers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2017 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-100781-5 (print) ISBN: 978-0-08-100782-2 (online) For information on all Woodhead Publishing publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisition Editor: Gwen Jones Editorial Project Manager: Charlotte Rowley Production Project Manager: Omer Mukthar Cover Designer: Alan Studholme Typeset by MPS Limited, Chennai, India

This book is dedicated to our families, with all of our love.

To my mother Taherh, and my father Mahmoud. Amir K. Shojaei

List of contributors

Liang Chen University of Lille 1, Villeneuve d’Ascq, France Zuorong Chen CSIRO Energy, Newcastle, NSW, Australia Jean-Baptiste Colliat University of Lille, Lille, France Gilles Duveau University of Lille 1, Villeneuve d’Ascq, France A. Ghassemi University of Oklahoma, Norman, OK, United States Richard Giot University of Lorraine/CNRS/CREGU, Vandœuvre-le`s-Nancy, France Albert Giraud University of Lorraine/CNRS/CREGU, Vandœuvre-le`s-Nancy, France Bezalel C. Haimson University of Wisconsin-Madison, Madison, WI, United States Paul Hauseux University of Lille, Lille, France Dawei Hu Institution of Rock and Soil Mechanics, Chinese Academy of Sciences, Beijing, China Rob Jeffrey SCT Operations Pty Ltd, Wollongong, NSW, Australia Alexandre Lavrov SINTEF Petroleum Research, Trondheim, Norway Xiaodong Ma University of Wisconsin-Madison, Madison, States; Stanford University, Stanford, CA, United States Moustafa Masri University of Lille 1, Villeneuve d’Ascq, France Panos Papanastasiou University of Cyprus, Nicosia, Cyprus Emmanuel Roubin University of Grenoble, Grenoble, France

WI,

United

xii

List of contributors

John W. Rudnicki Northwestern University, Evanston, IL, United States Ernestos Sarris University of Cyprus, Nicosia, Cyprus; University of Nicosia, Nicosia, Cyprus Jianfu Shao University of Lille 1, Villeneuve d’Ascq, France; Laboratory of Mechanics of Lille, Villeneuve d’Ascq, France; Lille University of Science and Technology, Villeneuve d’Ascq, France Wanqing Shen Lille University of Science and Technology, Villeneuve d’Ascq, France Amir K. Shojaei DuPont Performance Materials, DuPont, Wilmington, DE, United States George Z. Voyiadjis Louisiana State University, Baton Rouge, LA, United States Shouyi Xie University of Lille 1, Villeneuve d’Ascq, France Fan Zhang Hubei University of Technologies, Wuhan, China Xi Zhang CSIRO Energy, Newcastle, NSW, Australia Zhihong Zhao Tsinghua University, Beijing, China Hui Zhou Institution of Rock and Soil Mechanics, Chinese Academy of Sciences, Beijing, China

Preface

The focus of the proposed volume is on the Fracture Mechanics of porous rocks and modern simulation techniques for progressive quasi-static and dynamic fractures. The envisioned topics for this volume are defined to include a wide range of academic and industrial applications including petroleum, mining, and civil engineering. Chapters will focus on advanced topics in the field of rock’s fracture mechanics and address theoretical concepts, experimental characterization, numerical simulation techniques, and their applications as appropriate. As seen in the table of contents, “rock failure mechanics” is defined relatively broadly and includes a number of techniques and applications. Each chapter is intended to reflect the current state-of-the-art in terms of the modern use of fracture simulation in industrial and academic sectors. The authors of chapters have adhered to a “high-level” outline template that ensures continuity across the volume. Some of the major contributions of this volume include, but are not limited to: G

G

G

G

G

Anisotropic elastoplastic deformation mechanisms in fluid saturated porous rocks; Dynamics of fluids transport in fractured rocks and simulation techniques; Fracture mechanics and simulation techniques in porous rocks; Fluid structure interaction in hydraulic driven fractures; Advanced numerical techniques for simulation of progressive fracture in porous rocks including cohesive zone models, discreet elements approach, anisotropic porous rock’s continuum damage mechanics, extended finite element, and Enriched Finite Element; Multiscale modeling and micromechanical approaches for porous rocks; and Quasi-static versus dynamic fractures in porous rocks. G

G

G

G

G

G

G

During past few years, there has been a considerable proliferation of journal articles in the field of numerical and experimental studies of porous rock’s failure mechanisms. Often, this interest is driven by the oil and gas industries where hydraulic fracturing and drilling efficiency optimization tasks play a crucial role in the cost reduction. Nonetheless, there are few clear repositories of porous rock’s fracture mechanics; and most of the existing text books on this topic do not cover recent advances in this field. Thus, the proposed volume aims to fill a significant gap in the available literature and should serve as an important resource for petroleum, geomechanics, mining, civil, and structural engineers in industry as well as graduate students and professors in academy. Amir K. Shojaei and Jianfu Shao December 2016

Introduction

Rock mechanics is a branch of mechanics that studies the response of rocks to the external thermomechanical loads. The theoretical and applied science of rock’s mechanical behaviors have been utilized in: G

G

G

Structural engineering in which the stability of engineering structures are studied, Mining in which fracture of rocks in drilling, blasting, cutting, and grinding is investigated, Oil and gas applications, which are concerned with mechanical behavior of rocks in hydraulic fracturing and drilling processes.

Stability of rock structures subjected to wind loads or earthquake random vibrations is one of the design criteria in structural engineering. The energy changes induced by mining in the rocks surrounding large excavations may cause rock bursts. On the other hand wellbore stability in drilled oil wells is closely related to the fractured rock hydromechanics. The fluid content and fluid boundary conditions also affect the mechanical behavior of porous rocks in which governing equations for saturated and unsaturated rocks have been extensively studied in the literature. Due to inherent complexities in rock fracture mechanics, the question that still remains unanswered in—what knowledge, theory, and analytical procedures may be sufficient to justify the formulation or recognition of a coherent, dedicated rock fracture mechanics discipline? This book is dedicated to discuss the most promising engineering methods for such complex analysis. While chapters are designed to provide detailed information on each topic, in the following some brief discussion on the nature and content of the rocks fracture mechanics discipline are provided.

Rocks fracture mechanics Classical fracture mechanics is basically developed for conventional engineering material in which a tensile stress field causes a crack. Sophisticated theories have been developed for this type of fracture. On the other hand the fracture mechanisms in porous rocks differ from conventional materials. The stress fields operating in rock structures are pervasively compressive, and the mechanical behaviors of rocks are affected by the fluid mechanics. In many cases, such as hydraulic fracturing, the fracture is driven by the fluid rather than mechanical loads. So the established classical fracture mechanics theories might not immediately applicable to the fracture of rock. It must be emphasized that fracture mechanics approaches are only strictly correct when applied to the fracture initiation under static stress conditions and also

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only fracture initiation can be predicted by these methods. Once fracture has initiated, propagation of the cracks is a relatively complex process and it is largely fortuitous if fracture mechanics methods can be so successfully applied to the prediction of the crack path. In the case of porous rocks, the fracture propagation is largely controlled by the frictional sliding mechanisms. So theories that take into account compressive stress fields and friction forces, might successfully predict the fracture propagation path. However, fluid mechanics effects have not been incorporated in the classical frictional based fracture theories. A more comprehensive simulation technique for incorporating the physics of the fracture mechanisms in porous rocks is needed.

Scale effects on fracture behavior When dealing with large scales, such as rock behavior in drilling processes, most of the classical rocks fracture mechanics treat the rock body as a homogeneous, isotropic solid. This assumption ignores the effect of major geological discontinuities. The immediate deficiency of such approach is obvious to the reader when these methods are applied to the fractured and geologically discontinuous rocks within the earth’s shallow crust. Joints, natural fractures, and other fractures of geological origin are ubiquitous features in a body of rock. The discontinuous nature of rock masses may profoundly affect rocks’ mechanical and fracture behavior. Another size effect issue is encountered when the scale of observation is reduced to microscale in which rocks’ heterogeneous microstructures needs to be considered. At the microscale, deformation and fracture mechanisms of porous rocks are governed by their microstructure, which results in anisotropic deformation processes. The size effect in microscale is affected by, (1) the solid skeleton geometry and properties, (2) porosity, void shape, and patterns, and (3) presence of faults and fractures in the rock body. So, the rock properties together with the various structural geological features govern the mechanical behaviors and the scale of rock sample is an important feature in rock fracture mechanics studies.

Effect of tensile and compressive stress fields on rocks’ fracture mechanics Rocks exhibit little resistance to tensile stress fields, and their low tensile strength is distinguished from most of common engineering materials. Rock material specimens tested in uniaxial tension may fail at stresses an order of magnitude lower than compression strengths. Presence of voids at small scale and joints and other fractures in large scales results in low resistance to tensile stresses. In many text books rocks are assumed as a “no-tension” material, meaning that tensile stresses cannot be generated or sustained in a rock mass. Identifying zones that might be subjected to tensile stress fields is critical when designing for an excavation or oil well drilling. The energy release due to tensile rock failure may cause localized instabilities.

Introduction

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Rock fracture mechanisms and fluid effects Stability and fracture responses of a rock structure can be affected by the fluid mechanics. The fluid pressure within the rock skeleton induces tensile stresses, which reduce the compressive stress acting across a fissure or fracture plane. Hence the frictional resistance that causes interlocking of blocks of rocks can be reduced. Fluid effects on rocks’ fracture mechanics can be studied from different perspectives. The most obvious effect is through the operation of the effective stress law in which pore pressure affect the stresses within rock solid skeleton. In the case of hydraulic-driven fractures, coupling between fluid mechanics and mechanical responses of rock’s solid skeleton govern the fracture initiation and propagation. Due to these fluid-driven fracture mechanisms, the fracture toughness and behavior of the rock sample, observed under fluid-driven fractures, may differ from fracture responses measured via classical fracture mechanics tests. Another important effect of fluids on fracture of large rocks can be seen in saturated joints in which the shear strength is considerably reduced. Joints are usually under compressive stress fields and fluids under pressure lower the shear strength between two rock surfaces, which can be mobilized by friction. Another subtle fluid effect on rock fracture mechanics arise from the deleterious action of water on rocks and minerals. Mechanical properties of many rock samples, such as clay seams, are softened in the presence of groundwater resulting in lower strength. So, mechanical properties of rocks are determined by the geohydrological environment, and strength properties of rock samples under variable fluid conditions need to be predicted.

Environmental effects Although physical processes such as thermal cycling and weathering may be important near the surface, underground environmental effects are chiefly chemical in origin. The environmental effects may include dissolution and ion exchange phenomena, oxidation, and hydration. Analogous to corrosion effects on conventional materials, rocks’ fracture properties can be altered via reaction with gas and aqueous solutions. Chemical attacks may result in mechanical properties deterioration and influence the fracture properties of the intact rocks. The chemically and physically altered rock surfaces exhibit significant effects on the coefficient of friction between rock surfaces. Since the coefficient of friction governs frictional sliding mechanisms between rock layers, the chemical attach effects might be significant on fracture behaviors. It is clear that the fracture mechanics of rocks must include a number of topics that are not of concern in any other engineering discipline. The influence of temperature upon the strength of rock is significant at great depths where the temperatures approach the melting point of the rock constituents. The reduction in strength, stiffness, and fracture toughness could be of importance to drilling engineers.

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Time effect on rocks’ failure behavior Time influences deformation and fracture mechanisms in rocks. Time effects involve the deformation or fracture of rock under conditions of prolonged constant applied stresses, e.g., creep, or high rate loadings, e.g., perforation. Time-dependent mechanical behavior of rock has been intensively investigated, both theoretically and experimentally. For the design engineer who faces with the problem of designing a drilling or excavation job, these researches are barely incorporated in the design scopes. Most common practice for dealing with time effects is to use the results of short time or quasi-static laboratory tests with a liberal allowance for the possible strength alternation with time. These methods may not be always applicable because the extent to which the fracture behavior will change with time depends upon so many unknown factors that no definite rules could be suggested. A more realistic computational methodology that considers most of the physical mechanisms is needed to effectively predict the time effects. Amir K. Shojaei

Application of rock failure simulation in design optimization of the hydraulic fracturing

1

A. Ghassemi University of Oklahoma, Norman, OK, United States

1.1

Introduction

Extraction of unconventional geothermal and petroleum resources is made possible by reservoir stimulation to provide access a large volume of rock with a network of fractures for fluid flow and/or heat exchange. For petroleum resources development, stimulation is often accomplished by multiple hydraulic fracturing of horizontal wells, whereas stimulation of enhanced geothermal systems has usually relied upon injectioninduced shear slip on preexisting fractures and their coalescence. Maximizing facture permeability over the reservoir lifetime is the desired goal, and thus modeling and analysis of the stimulation process are called upon to guide the practice. The hydraulic fracturing process presents a complex mathematical problem that involves the mechanical interaction of multiple propagating fractures, fluid diffusion into the rock mass, heat transfer between the fluid and the rock, and hydraulic fracture interactions with natural fractures. Therefore, application of rock fracture mechanics and advanced modeling are necessary for improved understanding of the process and its optimization. Numerical modeling can be used to predict fractures trajectories, dimensions, and the induced stresses in the reservoir for the interpretation of seismicity and refraction analysis. This paper presents an overview of hydraulic fracturing conceptual models and rock failure mechanisms followed by a series of numerical simulations and analyses to provide some insight into various phenomena observed in hydraulic fracturing of unconventional reservoirs. In particular, two-dimensional (2D) and three-dimensional (3D) examples are used to illustrate the impact of hydraulic fracture interactions with each other and with discontinuities and highlight the role of rock anisotropy and heterogeneity and mixed-mode stimulation in relation to permeability change.

1.2

Reservoir stimulation by hydraulic fracturing of horizontal wells

Horizontal well stimulation usually involves creating multiple fractures along the wellbore using different well completion techniques. These multiple fractures Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00001-4 © 2017 Elsevier Ltd. All rights reserved.

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Porous Rock Failure Mechanics

generate large contact areas with the reservoir and also increase reservoir permeability. The multistage fracturing of a single or multiple horizontal wells is usually carried out either in simultaneous or sequential manner. In simultaneous fracturing, the multiple fractures are initiated and propagated at the same time (Fig. 1.1), whereas in sequential fracturing the fractures are created from one cluster after another in a well, usually by keeping the previously created fractures either propped or pressurized with fluid (Rodrigues et al., 2007; Soliman et al., 2008). Simultaneous or sequential fracturing is carried out in multiple wells and is called the “zipper” fracturing technique. The “zipper-frac” approach aims to create a network of long and closely spaced hydraulic fractures. Zipper fracturing also has operational benefits when multiple wells are stimulated from the same pad. Instead of simultaneous fracturing of parallel wells, stimulation can also be performed in a sequential manner that is called sequential zipper fracturing. In this method, an adjacent lateral is stimulated while restricting flow back from the stimulated fractures in another lateral. Fig. 1.1 illustrates the difference in stimulation design between simultaneous and sequential zipper fracturing. Over 50% of current shale plays employ zipper fracturing technique for operational benefits (Jacobs, 2014). In sequential stimulation of a single well a few hours are spent on wireline operations (setting plugs and perforating) before continuing on to the next stage. In sequential zipper fracturing, these operations can be performed on a neighboring lateral while stimulating is done on another lateral. Even though most operators have adopted the zipper fracturing technique for its operational efficiency, improved production has been observed in a few cases (Jacobs, 2014). A modification of “zipper” fracturing termed “modified zipper” fracturing (MZF) also has been discussed (Rafiee et al., 2012) based on stress analysis. In the MZF method, the stimulation stages in neighboring laterals have an initial offset between them. Sesetty and Ghassemi (2015) investigated the total stimulated rock volume by fracture networks obtained from conventional and MZF and concluded that theoretically, both methods of fracturing yielded almost equal stimulated reservoir volume. However, it was pointed out that in MZF, fracture turning due to stress shadow effects in the overlap region is more conducive to reactivation of preexisting natural fractures, potentially increasing complexity and the stimulated volume. Stresses changes in the vicinity of hydraulic fractures mainly depend on the net fluid pressure inside the fracture and the fracture geometry (Warpinski and Branagan, 1989; Ge and Ghassemi, 2008; Ghassemi et al., 2013; Safari and Ghassemi, 2014). These induced stresses in the region surrounding the hydraulic fractures are termed the “stress shadow.” The effect is a more dominating in the case of multiple closely-spaced fractures in which mechanical interactions among fractures may restrict or terminate propagation of some of the fractures (El Rabba, 1989). The stress shadowing could lead to reduction of fracture aperture, increase the risk of proppant screen-out, and fracture reorientation due to altered stress conditions. The effect of stress shadowing when closely spaced multiple hydraulic fractures are created parallel to each other is of major interest for numerical simulation of multistage fracturing. The spacing between the fracture surfaces, net fluid

Application of rock failure simulation in design optimization of the hydraulic fracturing σV

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Figure 1.1 Schematics of (A) simultaneous fracturing (single stage) of a lateral, (B) conventional zipper fracturing of multiple wells, (C) modified zipper fracturing of multiple horizontal wells. Stages are performed from toe to heel of a lateral. Note that figures are not depicting any interactions or stress effects. The fracturing can be carried out simultaneously or sequentially as depicted in the 2D images. Perforations are created in the direction N
S (parallel to maximum principal stress direction).

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pressure, and in-situ stress contrast (i.e., the difference of the maximum and minimum horizontal stresses) play an important role in the mechanical interaction between the fractures (Tarasovs and Ghassemi, 2014; Sesetty and Ghassemi, 2015; Wong et al., 2013). Therefore, optimization of the fracture spacing is critical in the multistage fracturing, both from technical and economical point of view, because it affects both fractures creation and subsequent well productivity. A fully coupled numerical model capable of simulating multistage fracturing can be a vital tool in understanding the effects of various parameters such as well spacing, fracture spacing, and fracture offset. The numerical models need to consider the relevant rock failure and fracture propagation mechanisms involved in the hydraulic fracturing concepts.

1.3

Hydraulic fracturing conceptual models

Hydraulic fracturing numerical models have been developed on the basis of specific conceptual models. Early hydraulic fracturing numerical models (e.g., Carter et al., 2000; Ouyang et al., 1997; Yew, 1997; Lee et al., 1994; Clifton and Wang, 1991; Clifton and Abou-Sayed, 1981; Vandamme and Curran, 1989; Cleary and Wong, 1985; Wiles and Curran, 1982; Abe et al., 1976) were developed for conventional reservoirs based on the concept of a tensile fracture propagating under fluid pressure (e.g., Barenblatt, 1962; Perkins and Kern, 1961; Geertsma and de Klerk, 1969). However, field observations of micro seismicity during fracturing of geothermal reservoirs (e.g., Cornet et al., 1997; Roff et al., 1996; Fehler, 1990) and surface-area estimations based on gas production in shale reservoirs (e.g., Ghassemi and Suarez-Rivera, 2012) show the existence of a complex fracture network, suggesting shear-slippage as a major mechanism of stimulation (“hydro-shear”). Many hydraulic fracturing jobs have shown shear failure to be a major source of permeability enhancement, particularly where natural fractures are pervasive (Rutledge et al., 1998; Mayerhofer et al., 1997; Chipperfield et al., 2007). Different interpretations of field experiences have spurred different stimulations concepts and numerical models for their analysis. However, for the relatively high injection rates (50
80 bbls/min) and pressures, one can expect a major hydraulic fracture to initiate and extend in the tensile mode. Leakoff from the major hydraulic fracture into the rock would cause slip on preexisting or newly formed discontinuities and enhances the permeability in the vicinity of the main fracture via shear stimulation. Ghassemi et al. (2013) studied the three-dimensional stress and pore pressure distributions around a hydraulic fracture and analyze the potential for formation failure resulting from pressurization. Simulation results for constant water injection into a rectangular fracture in Barnett Shale showed that rock failure can occur in the vicinity of the fracture especially near the fracture tips. The dominant failure mode in the rock matrix was tension in the close vicinity of the fracture (1
2 m off the fracture walls) where the pore pressure attained its highest values. Shear failure potential was observed to exist away from the fracture walls where shear stresses

Application of rock failure simulation in design optimization of the hydraulic fracturing

7

are sufficiently high to overcome the strength of the rock. The shear failure of the intact rock near the main hydraulic fracture is less likely for many of the strong source rocks (the compressive strength of some mudstones equals or exceeds that of Westerly granite). Therefore, reactivation and propagation of preexisting cracks would be the main source of permeability increase.

1.3.1 Rock failure and the stimulated volume The “hydro-shear” conceptual model envisions injecting into a rock mass at pressures below the minimum in-situ stress to cause slip on critically stressed fractures, and or shear failure of the rock mass. On the other hand, based on a detailed review of a number of enhanced geothermal system experiments, it has been suggested (Jung, 2013) that “hydro-shear” conceptual model is not an effective approach for geothermal reservoir development. Instead, the formation of tensile “wing cracks” has been advocated as the mechanism of stimulation. This has led some to believe that the hydro-shear concept is not relevant to reservoir stimulation. However, wing cracks (and shear dilation) are in fact, an integral part of the shear-slip stimulation mechanism, and shear propagation is also plausible and can contribute to permeability and micro-seismicity. This is similar to the process of shear failure in laboratory triaxial compression tests on rock whereby tensile cracks coalesce to form a shear crack. Although individual tensile cracks do form, the failure is shear. Formation and extension of wing cracks in dry closed cracks has been studied analytically, numerically and experimentally under different loading conditions. Extensive research has been carried out on the propagation of wing cracks including the early works of Bombolakis (1973), Hoek and Bieniawski (1984), Horii and Nemat-Nasser (1986). Both of Bombolakis (1973) and Horii and Nemat-Nasser (1986) conducted experiments on open flaws in C.R.39 photoelastic plates. Hoek and Bieniawski (1984) studied the failure of open cracks in glass samples under compression. The study of wing cracks by Petit and Barquins (1988) has shown that shear failure as fundamental failure mechanism is tenable in brittle rocks. As Rao et al. (2003) argue, under compression-shear loading condition, mode I stress intensity factor often reaches its maximum before the mode II stress intensity factor, and since Mode I toughness, KIC, is generally lower than that of mode II, KIIC, cracks propagate in mode I (wing crack) first. However, it is important to note that the shear stress concentration in the vicinity of the kink point may exceed the shear strength of the material by further shearing on the crack surfaces giving rise to Mode II crack propagation. In fact, experimental (Petit and Barquins, 1988) and numerical results (Shen and Stephansson 1994; Bobet and Einstein, 1998) indicate that Mode II propagation can occur along with mode I propagation in closed cracks under compression-shear loading condition. The analysis of closed cracks is challenging as there is not any widely accepted failure criteria for such cracks. Furthermore, the presence of friction and dilatancy gives rise to the complexity of closed crack analysis (Scavia, 1995). The initial crack remains mechanically closed under compression, whereas the wing cracks are open throughout the loading. Formation of secondary cracks that are in-plane with the preexisting cracks

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σyy Inclination angle Injection point σxx

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Figure 1.2 Schematic of crack geometry and in-situ stresses.

has been reported in several works (Shen and Stephansson, 1994; Bobet and Einstein, 1998; Vesarhelyi and Bobet, 2000; Li et al., 2005; Cao et al., 2015). These in-plane secondary cracks form at the tips of the flaws as shear fractures. Reservoir stimulation by shear slip and the formation of secondary cracks has been recently studied by Kamali and Ghassmei (2016a, b). Using a series of numerical simulations of isothermal injection in the center of a closed natural fracture in impermeable rock (Fig. 1.2), it was shown that that both Mode I and II can occur as a result of injection at or below the minimum in-situ stress. Fluid injection reduces the effective normal stress on the natural fracture, lowering the fracture closure, and increasing its opening. Further injection may cause slip on the fracture which, in turn, results in wing crack initiation and propagation. It was shown that shear propagation (mode II) also occurs once the shear stress at the kink point exceeds intact rock shear strength. Propagation of the secondary cracks was considered based on the equivalent stress intensity factor for mixed-mode propagation, and shear failure in the rock matrix along the preexisting fracture. Fig. 1.3 shows the resulting wing cracks for different levels of differential stress. The total shear deformation observed at the injection point is also presented in Fig. 1.1. It can be observed that the total shear slip starts to increase once the effective normal stress is reduced to a critical value due to injection. Additionally, the amount of total shear displacement is higher when the differential stress is higher. Note the difference in the pressure-time profile at the injection point for these cases. It can be seen in Fig. 1.3 that for the case of lowest differential stress, the pressure nearly increases to the minimum principal stress before the propagation begins. The pressure remains constant as the crack propagates. On the other hand, natural fracture propagation begins at pressures lower than the minimum in-situ stress for the cases with higher initial differential stress. Simulations also show that a higher differential stress and confining pressure favors propagation of shear cracks. Shear propagation begins once the local shear stress at the tip of the preexisting natural fracture exceeds the local shear strength. The mode of propagation is greatly impacted by the cohesion of the intact rock. It can be observed in Fig. 1.4 that both shear and wing cracks form, and the length of the shear crack is almost equal to

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Figure 1.3 Wing crack trajectory from a 45-degree natural fracture under a constant confining stress of 35 MPa. Shear displacement and the injection pressure profile at the injection point for different differential stresses at a constant confining stress. Intact rock cohesion was 15 MPa. See Kamali and Ghassemi (2016a,b) for other parameters used in this example.

that of wing cracks. The length of the shear crack decreases as the cohesion increases because the higher shear strength at the crack tip prevents shear propagation. It is interesting to note that shear crack propagates in a plane approximately parallel to the plane of the preexisting natural fracture. The pressure profile for this example is shown in Fig. 1.5. The pressure
time profile at the injection point almost coincides for c0 5 12 MPa and c0 5 15 MPa. The insignificant difference in the pressure profile may be attributed to the fact that the shear fracture segment has a significantly smaller aperture as compared to the wing crack segment. Therefore, most of the injected volume is taken by the wing cracks, but the shear fractures do contribute to creation of new surface area.

1.4

Mechanical interactions of multiple hydraulic fractures

Stress interference between cavities and fractures in rock are well known from elasticity considerations. Field and laboratory observations (e.g., El Rabba, 1989; Waters et al., 2009) also indicate stress alterations by hydraulic fractures. The

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Porous Rock Failure Mechanics

Aperture (mm)

C = 12 MPa

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12

1

Y

0.5 0

–0.5 –1 –1.5 –1.5

1.5

0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12

0.5 0

–0.5 –1 –1.5

–1

–0.5

0

0.5

1

1.5

–1

0

X

1

X

Aperture (mm)

0.24 0.22 0.2 0.18 0.16 0.14 0.12

1

0

1.5

phi = 35 deg

Aperture (mm)

0.24 0.22 0.2 0.18 0.16 0.14 0.12

1 0.5 Y

phi = 30 deg

Y

Aperture (mm)

C = 15 MPa

1

Y

1.5

0

–0.5 –1

–1 –1

0 X

1

–1.5 –1.5

–1

–0.5

0

0.5

1

1.5

X

Figure 1.4 A 60-degree natural fracture subjected to 20 MPa and 30 MPa of in-situ stresses with the cohesion of the intact rock in the range of 12
15 MPa. Wing crack and shear propagation for different rock cohesion values (A), (B) Wing crack and shear fracture trajectories for different values of intact rock friction angle (Kamali and Ghassemi, 2016a,b).

objective of the multiple hydraulic fracture design is maximizing well access to the reservoir while minimizing costs (water and proppant usage, equipment time, and pumping energy). This can be achieved by analyzing the hydraulic fracturing process to avoid excessive interactions between the fracturing stages. The following examples illustrate some of the fundamental rock mechanics issues involved using 2D and 3D numerical simulations of multiple fracture propagation.

1.4.1 Some insights on hydraulic fracture spacing optimization using numerical simulations For a given reservoir rock, the stress shadow, in-situ stress contrast, and the injection conditions impact the fracturing outcome. The in-situ stress and stress

Application of rock failure simulation in design optimization of the hydraulic fracturing

11

25

Pressure (MPa)

20

15

10 C = 12 MPa

5

0

C = 15 MPa

0

20

40

60

80

100

120

Injection time (s)

Figure 1.5 Pressure
time profile at the injection point for three intact rock cohesion values.

Input parameters form the Niobrara chalk (Maldonado et al., 2011; Hope et al., 2013)

Table 1.1

Parameter

Value

Formation depth (h) Young’s modulus (E) Poisson’s ratio (ν) Mode-I fracture toughness (KIC) Vertical stress (σV) Minimum horizontal stress (σh) Maximum horizontal stress (σH)

2316.5 m 34.1 GPa 0.29 2.0 MPa m0.5 52.4 MPa 38.0 MPa 43.5 MPa

shadow effects have been illustrated in 3D by Kumar and Ghassemi (2016) by considering the 3D problem of two initially parallel transverse fractures growing in an anisotropic in-situ stresses field. The numerical model has been developed using a combination of the boundary element method and the finite-element method. Fracture propagation is considered within the framework of the linear elastic fracture mechanics. The fluid flow inside the fracture is assumed to be laminar, and the fluid is Newtonian. The Galerkin’s finite-element approach is used for fluid-flow modeling; the rock mass deformation is simulated using the elastic displacement discontinuity method (DDM). The crack tip displacement approach is implemented for mixed-mode fracture propagation (see Kumar and Ghassemi, 2016 for details). The relevant input data in the simulation shown here are listed in Table 1.1. A uniform pressure of 55 MPa is applied over the fracture surfaces.

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Porous Rock Failure Mechanics Z

Minimum horizontal stress distribution

(A) 1

X

Z (m)

2

–2

–2

Y

Toe

(m

0

)

Heel 2

–4 –2

0 2 m) X (m

4

Y w (mm) 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6

(B)

σxx(MPa)

5

Y (m)

4

0

2

1

2

54 52 50 48 46 44 42 40 38 36 34 32

0

–5

–5

0

5

10

X (m) Z Minimum horizontal stress distribution

1

X

2 4

0

–2

–4 –2 Y (m 0 m) 2

Z (m)

2

Toe Heel 4

6

4

2

0

m)

X (m

–4 –2

3.6 3.2 3.4 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6

2

1

Y w (mm)

(D) σxx(MPa)

5

Y (m)

(C)

54 52 50 48 46 44 42 40 38 36 34 32

0

–5

–5

0

5 X (m)

10

Figure 1.6 Distributions of fracture geometries and aperture and the minimum horizontal stress (in the central plane perpendicular to the fractures) for a simulation time of 13.55 min. (A and B): spacing 5 3 m, (C and D): spacing 5 6 m (after Kumar and Ghassemi, 2016).

The distributions of fractures opening, geometries, and changes in the minimum horizontal stresses in the x
y plane are shown in Fig. 1.6 for a fracture spacing of 3 m and 6 m, after 20 propagation steps. It can be seen that as the spacing between the fractures increases, the intensity of the stress shadow decreases and less curving, and higher fracture openings are observed. The areal distribution of the stress shadow zone would vary with the fracture size and geometry, and rock anisotropy (see the next section). As a second example, consider the effect of horizontal stress anisotropy on the multiple fracture propagation. Two initially parallel fractures spaced at 3 m are subjected to a uniform constant fluid pressure of 55 MPa. Fig. 1.7 shows the resulting distributions of fracture opening, propagated geometry, and the change in the maximum horizontal stress for two cases of horizontal stress differential, (σH
σh), of 1 MPa and 10 MPa. After 20 propagation steps, the fractures have experienced a greater curvature in the case of low stress differential, so that for strong stress anisotropy the stress shadow effect is not strongly dominant. Hence, for multiple

Application of rock failure simulation in design optimization of the hydraulic fracturing Z

(A) 1

Minimum horizontal stress distribution 1

X Y

2

13

(B)

2

σxx(MPa)

5

54 52 50 48 46 44 42 40 38 36 34 32

w(mm)

Z (m)

0

–2 –2

Toe

Y

0

(m

Heel

–2

)

2

2

4

Y (m)

2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6

2

0

–5

0 )

–5

Z

1

X

2

1

0

Z (m)

2

–4 –2 Toe )

2 4

4

2

0 ) X (m

–4 –2

(D)

2

Y

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6

σxx(MPa)

5

Y (m)

4

Heel

10

Minimum horizontal stress distribution

w(mm)

0

5 X (m)

(C)

–2 Y (m

0

X (m

54 52 50 48 46 44 42 40 38 36 34 32

0

–5

–5

0

5

10

X (m)

Figure 1.7 Distributions of fractures opening, geometry, and the minimum horizontal stress (in the central plane perpendicular to the fractures) after 20 propagation steps (simulation time of 14.13 min): (A and B) stress anisotropy 5 1 MPa, (C and D) stress anisotropy 5 10 MPa (after Kumar and Ghassemi, 2016).

fracturing in typical unconventional reservoir rocks, and a given injection condition, increased stress anisotropy tends to mitigate the curving induced by the mechanical interaction of the fractures. The mechanical interactions between the hydraulic fractures have also been observed in field experiments. For example Waters et al. (2009) describe a field study involving the simultaneous (zipper) fracturing of multilaterals in the Woodford shale, detailing preferential growth in the Stage 6 of the stimulation. There were four parallel treatment wells with a monitoring well on each side of the treatment zone. The stimulation job was carried out on all four laterals simultaneously in seven stages. Each stage had four perforation clusters 38 m (125 ft) apart, and slick water was injected at 80 bpm. Observations from the micro-seismic plot showed that most of the seismic activity occurred towards the heel sections of most stages, and a few perforation clusters in between the stages were left unstimulated. This behavior is illustrated in Fig. 1.8A where the simultaneous treatment of one and two wells is modeled (Sesetty and Ghassemi, 2016).

200

Isotropic Aperture (mm)

200

2.4 Stage-1

Stage-2

100

2.0

2.0

100

1.6

1.4 1.2

Y (m)

1.6 0

1.8

50

1.8 Y (m)

2.4 2.2

2.2

Stage-3

Aperture (mm)

Isotropic

150

Well-1

0

1.2

–50

Well-2

1.0 0.8

–100

0.6 Toe

Heel

–200

1.0 0.8

–100

0.6 –150

0.4

0.4 0.2

1.4

–200 –250

0

100

200

300 X (m)

400

500

600

0

100

200 X (m)

300

400

Figure 1.8 Left: the network of fractures from a single lateral stimulated in three stages. The outer fractures on the west side of Stages 2 and 3 are unstimulated; Right: the network obtained from simultaneous stimulation of two lateral wells in isotropic rock. The fracture tips rotate in the region between the wells due to induced shear stresses. The fractures from Well-1 grow towards the north and the fractures from Well-2 grow towards the south avoiding the region of overlap in between the wells (after Sesetty and Ghassemi, 2015, 2016).

Application of rock failure simulation in design optimization of the hydraulic fracturing

15

The numerical results for the fracture geometry and aperture distribution in different rock systems are shown after 900 s of injection (i.e., 300 s into each of the three stages). It can be seen that in the single-well sequential fracturing case, the newer fractures that are closest to the dominant fractures of the previous stages do not grow. When simultaneously fracturing two parallel wells (Fig. 1.8B), the fractures tend to grow outward, avoiding the area in between the wells. Numerical simulations indicate that the outward growth is accelerated after the fractures in the overlap region “feel” each other’s presence. This is consistent with the results of Stage 6 for Wells 1 and 2 as illustrated by the time series of micro-seismicity in Waters et al. (2009).

1.4.2 The role of rock fabric and structure 1.4.2.1 Role of rock anisotropy Micro-seismic monitoring of hydraulic fracturing in shale (e.g., Waters et al., 2009) has shown favorable propagation of hydraulic fractures in some fracturing stages. Such preferred growth can be attributed to the presence of natural fractures and rock anisotropy. Recently, Sesetty and Ghassemi (2016) developed the first multiple hydraulic fracture model that considers shale anisotropy and accounts for the fracture toughness anisotropy. The model is based on the 2D DDM and accounts for the 3D stress shadow effect using a 3D correction factor developed for anisotropic rock. The model has been applied to explain the phenomena observed in Waters et al. (2009) in the context of rock anisotropy. A single-stage fracturing with four perforation clusters spaced 40 m apart were simulated (Fig. 1.6). The simulations were carried out for different levels of rock anisotropy: an isotropic rock and 2 different types of orthotropic rocks (A and D). Rock D has fracture toughness anisotropy of degree 2. These rocks are considered to have strong material anisotropy with Emax/Emin 5 3 and while they have the same elastic moduli, their Emax orientations with respect to the global x-axis are different. In all cases, the minimum Young’s modulus is in the vertical direction (Ev or Ez) and the intermediate (Emin) and the maximum (Emax) Young’s moduli are in the horizontal plane as shown. Slick water is injected at 30 bpm into the horizontal wells (see Sesetty and Ghassemi, 2016 for full model description and input details). The fracture apertures and propagation patterns in the rock systems are shown in Fig. 1.9 for 300 s of injection. In all cases, the outer fractures tend to grow the most. The maximum suppression in growth of the inner fractures is observed in the orthotropic-A rock. This is because in this case, the stress shadow between the fractures has a larger areal extent. In the orthotropic-D rock, fracture toughness anisotropy promotes more inner region fracture growth (see Sesetty and Ghassemi, 2016). Also, the fractures are slightly tilted towards the direction of the minimum fracture toughness in agreement with field observations. Overall, the fracture geometries and apertures obtained in each rock system are different indicating that material properties anisotropy affects the resultant stress shadows and thus the pattern of fracture network development.

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Porous Rock Failure Mechanics

Orthotropic-A

Isotropic 100

0

–50 –100

Aperture (mm) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

50 Y (m)

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

50 Y (m)

100

Aperture (mm)

0 –50

–100 –50

0

50 100 X (m)

150

–50 200 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Y (m)

50 0

–50

50 100 X (m)

Isotropic-A 6

1

150

Aperture (mm)

2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4

150

Aperture (mm)

100 50 Y (m)

Isotropic-D

100

0

2

3 4

5 Well-1

0 –50

Well-2

–100 –150 –200

–100

–250 –50

0

50 X (m)

100

150

0

100

200 X (m)

300

400

Figure 1.9 Top row: a single-stage stimulation with four fractures showing two outer dominant fractures and two center fractures with hindered growth. In orthotropic-A rock the outer fractures tend to impede the growth of the center fractures, whereas in orthotropic-D rock the center fractures show a higher growth due to toughness anisotropy. Compare the right figure in the last row to Fig. 1.8B. Anisotropy has increased the level of asymmetric growth and has increased the retardation of interior fractures (after Sesetty and Ghassemi, 2016).

1.4.2.2 Influence of natural fractures on hydraulic fractures It has also been suggested by Waters et al. (2009) that the observed limited length growth in Stages 1
3 may have been due to the presence of a fault and or natural fractures. The interactions between natural and hydraulic fractures have been the subject of many theoretical, modeling, and experimental/analytical studies (e.g., Blanton, 1986; Chuprakov et al., 2010; Zhang and Jeffrey, 2006; Wu et al., 2004; Dobroskok and Ghassemi, 2004; Koshelev and Ghassemi, 2003a,b; Lam and Cleary, 1984; Thiercelin et al., 1987). These studies have shown that natural fracture/fault inclination and frictional characteristics, as well as their surface conditions are the most significant factors that influence hydraulic fracture trajectories. Fracture arrest or its continued propagation by a jog at the interface is influenced by the fracture trajectory and whether it is attracted or rejected by the natural

Application of rock failure simulation in design optimization of the hydraulic fracturing

17

Input parameters for simulations of hydraulic fracture and natural fractures interactions

Table 1.2

Young’s modulus Poisson’s ratio Tensile strength Cohesion Coefficient of friction NF normal stiffness NF shear stiffness NF maximum closure Injection rate Fluid viscosity HF and NF Fractures height

27 GPa 0.25 0 MPa 10 MPa 0.3 30 GPa/m 300 GPa/m 1 mm 40 bpm 1 cP 30 m

fracture. Analysis of the stress field induced by interaction of a hydraulic fracture and a discontinuity suggests that (a) stiff contact on a discontinuity is more favorable for coalescence and reinitiation than a slipping contact; (b) slip on a discontinuity allows opening of the fracture at the point of coalescence. Fracture opening at the point of coalescence for a stiff contact is 2.4 times less than that for a contact with 100 times greater compliance (Koshelev and Ghassemi, 2003b); (c) an angle of approach of 30 degrees and less engenders propagation through a discontinuity by a jog. Recently, Sesetty and Ghassemi (2016) carried out relatively large-scale simulations to illustrate the impact of natural fractures on simultaneous fracturing with reference to the field case reported by Waters et al. (2009). A single-stage fracturing with five clusters spaced at 20 m has been numerically simulated. Two cases are shown below, one with a large stress differential and the other with a small stress differential. The input data for these simulations are shown in Table 1.2. For the low differential case, the maximum and the minimum principal stresses are 60 MPa and 59 MPa, respectively. As can be seen in Fig. 1.10, the trajectories of hydraulic fractures are controlled by the mutual interactions between the HF and the NF fracture/faults resulting in one or two dominant fractures depending on the in-situ stress conditions. In the low differential stress case, the final geometry is a more complex. The higher confining pressure promotes irregular fracture paths and crossing of natural fractures and the emergence of one dominant hydraulic fracture. In contrast, in the high differential stress case, the low confining pressure promotes more fracture arrests. The result is less tortuous, but asymmetric hydraulic fracture growth with the emergence of two dominant fractures. For the low differential stress case, a higher normal stress and a lower shear stress act on the faults/natural fractures promoting crossing while for the higher stress differential case, a lower normal stresses (and slip) on the faults promote fracture arrest.

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Porous Rock Failure Mechanics

Isotropic

59 MPa

Isotropic

60 MPa

100

50

50

0

Wellbore

Y (m)

150

100 Y (m)

150

0

–50

–50

–100

–100

–150

Aperture (mm)

40 MPa

60 MPa

Wellbore

–150

Aperture (mm)

0.0 0.6 1.2 1.8 2.4 3.0

0.0 0.6 1.2 1.8 2.4 3.0

–200

–200 –50

50

0

100

X (m)

–50

0

50

100

X (m)

Figure 1.10 Simultaneous fracturing in the presence of natural fractures (Sesetty and Ghassemi, 2016). Left: high differential stress case, σH 5 60 MPa, σh 5 40 MPa; Right: low differential stress case, σH 5 60 MPa, σh 5 59 MPa.

1.4.3 The importance of 3D effects For investigation of the height growth and containment aspects of multiple fracturing in horizontal wells, 3D simulations are necessary. 3D simulation of multiple hydraulic fracturing has been the subject of only a few studies in the past several years because of the complexities involved in 3D modeling (e.g., Wong et al., 2013; Rungamornrat et al., 2005; Kumar and Ghassemi, 2015, 2016). Fully 3D modeling increases the computational effort so that most simulations are limited to small-scale geometries. Some illustrative cases are shown here to highlight important issues. For demonstration purposes, a relatively higher initial fracture radius equal to 5 m is used. First, consider a scenario where the in-situ stresses are not constant but vary with depth. The distance between the fractures is set to 10 m. The reservoir rock properties correspond to the Niobrara chalk formation as listed in Kumar and Ghassemi (2016). An in-situ stress gradient of 0.3 MPa/m is assumed. This high stress gradient is used to emphasize the fracture propagation behavior in a short simulation time. The total injection rate is 0.30 m3/s and the fluid viscosity is 0.005 Pa s. The frictional pressure-drop in the wellbore and perforation losses are not considered. Hence, the fluid flow rate is equally divided among three fractures. The distributions of fracture openings and geometries after 15 propagations steps are shown in Fig. 1.11. The lower regions of the fractures experience higher horizontal in-situ stress resulting in less vertical growth and lower fracture openings. It is observed that middle fracture’s growth is restricted by the strong stress shadowing effect from the exterior fractures (e.g., Frac.1 and Frac.3), whereas the latter turn outward and tend to grow in height. In the next 3D simulation case, the horizontal stress is higher above and below the “pay-zone.” The initial radius of the fractures is 5 m and the distance between the fractures is 10 m. The higher in-situ stress zones were considered at 8 m above

Application of rock failure simulation in design optimization of the hydraulic fracturing

19

Minimum horizontal stress distribution 30

Z

(A)

Y

1

w (mm) 8.1 6.9 10 5.7 4.6 3.4 0 2.2 1.0

(B) σxx(MPa)

Y (m)

Z (m)

–10

0

–10 –20

30

Heel –10

50.0 47.9 45.8 43.6 41.5 39.4 37.3 35.1 33.0

10

–10

Toe

10 (m 0 )

3

20

3

Y

2

1

X

2

0

10 ) X (m

20

–30 –20

–10

0

10 X (m)

20

30

40

Figure 1.11 (A) Distribution of fracture opening under a vertical gradient in σh (simulation time 5 19.63 min). (B) Distribution of the minimum stress component. Note that height growth of exterior fractures is enhanced due to stress shadow effects.

Z

(A) 1

2

3

Y

X

Minimum horizontal stress distribution 30 1

2

(B)

3

σxx(MPa)

20

Y

0

Toe

)

(m

Heel

–10 –20–10

0

10 X (m)

20

30

Y (m)

Z (m)

10

52 50 48 46 44 42 40 38 36 34 32

w (mm) 10 9.1 8.1 10 7.1 0 6.1 5.1 0 4.1 3.1 –10 2.1 1.1 –20 –10 –30 –20

–10

0

10 X (m)

20

30

40

Figure 1.12 Fracture propagation from a wellbore that is oriented 15 degrees from the minimum horizontal direction. (A) Fracture kinking and out-of-plane growth; (B) distribution of the minimum stress component.

and below the horizontal well. The distributions of the fracture opening and geometries are shown Fig. 1.12(B). As can be seen, the fractures propagate less in the high stress zones and demonstrate the tendency to grow in the low in-situ stress zones. The stress shadow promotes curving away of exterior fractures and suppresses the opening and growth of the middle fracture. Often the wellbore is not oriented along the σh, and this impacts fractures paths and final geometries. Consider the previous example but with σh deviated 15 degrees from the x-axis. In this case, the initial perforations are not orthogonal to the

20

Porous Rock Failure Mechanics

minimum principal stress direction (i.e., x-axis); so that the fractures tend to propagate in a nonplanar manner as they evolve to extend along the direction of the maximum horizontal stress in horizontal plane and along the direction of maximum principal stress (z-axis). This behavior increases fracture tortuosity and can cause large pressure drop neat the wellbore. It can also sever as choke point for proppants.

1.5

Conclusions

Hydraulic fracturing in shale is a multiscale and multiphysics process requiring simulation of a sizable rock mass subjected by fluid injection over as much as several hours or more. Shales or mudstones are characterized by a multiscale fabric, anisotropy, and heterogeneity that significantly increase the complexity of hydraulic fracturing of horizontal wells. The process involves propagation of multiple (as many as ten or more) interacting hydraulic fractures in a heterogeneous stress field coupled with poroelastic and at times thermoelastic influences. Micro-seismic activity accompanies hydraulic fracturing and is used to quantify the simulated volume and to assess the impact of reservoir stimulation. Interpretation of the seismicity indicates the prevalence of mixed-mode rock failure and fracture propagation. Rock mechanics and rock fracture mechanics have been applied to better understand the multiple hydraulic fracturing process and to predict the stimulation outcome; however, some questions remain outstanding and require further research to more effectively design multiple fracturing jobs. Many advanced numerical techniques have been proposed in the past few years, and a number or promising numerical models have been developed. However, their utility for field applications has been somewhat limited mainly because of the significant computational time needed for a moderate-scale field example. While research efforts will continue to remove existing barriers to understanding and to technology development, physics-based models that capture the main aspects of multiple hydraulic fracture propagation will continue to be valuable tools for guiding engineering design.

References Abe, H., Mura, T., Keer, L.M., 1976. Growth rate of a penny-shaped crack in hydraulic fracturing of rocks. J. Geophy. Res. 81 (29), 5335
5340. Barenblatt, G.I., 1962. The mathematical theory of equilibrium cracks in Brittle fracture. Adv. Appl. Mech. Vol. 7, 55
129. Blanton T.L. 1986. Propagation of hydraulically and dynamically induced fractures in naturally fractured reservoirs. Proc. Unconventional Gas Technology Symp. Of the SPE. 613
621. Louisville. Bobet, A., Einstein, H.H., 1998. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. Int. J. Rock Mech. Min. Sci. 35 (7), 863
888. Bombolakis, E.G., 1973. Study of the brittle fracture process under uniaxial compression. Tectonophysics. 18 (3
4), 231
248.

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Cao, P., Liu, T., Pu, C., et al., 2015. Crack propagation and coalescence of brittle rock-like specimens with pre-existing cracks in compression. Eng. Geol. 187, 113
121. Carter, B.J., Desroches, J., Ingraffea, A.R., Wawrzynek, P.A., 2000. Simulating fully 3D hydraulic fracturing. Model. Geomec. 525
557. Chipperfield, S.T., Wong, J.R., Warner, D.S. et al., 2007. Shear Dilation Diagnostics: A New Approach for Evaluating Tight Gas Stimulation Treatments. Paper presented at the SPE Hydraulic Fracturing Technology Conference, College Station, Texas USA. Society of Petroleum Engineers 106. Chuprakov, D.A., Akulich, A., Siebrits, E., Thiercelin, M., 2010. Hydraulic Fracture Propagation in a Naturally Fractured Reservoir. Paper presented at the SPE Oil and Gas India Conference and Exhibition, Mumbai, India. SPE 128715-MS. Available from: http://dx.doi.org/10.2118/128715-MS. Cleary, M.P., Wong, S.K., 1985. Numerical simulation of unsteady fluid flow and propagation of a circular hydraulic fracture. Int. J. Num. Anal. Methods Geomech. 9 (1), 1
14. Clifton, R.J., Abou-Sayed, A.S., 1981. Variational Approach to the Prediction of the Three Dimensional Geometry of Hydraulic Fractures, in: SPE/DOE 9879, Low permeability Gas Reservoir Symp. Denver, CO. Clifton, R.J., and J.J., Wang. 1991. Modeling of Poroelastic Effects in Hydraulic Fracturing. SPE 218171, Rocky Mountain Regional Meeting and Low Permeability Reservoirs Symposium, Denver, CO: 661
670. Cornet, F.H., Helm, J., Poitrenaud, H., Etchecopar, A., 1997. Seismic and aseismic slips induced by large-scale fluid injection. Pure Appl. Geophys. 150, 563
583. Dobroskok, A., Ghassemi, A., 2004. Crack propagation, coalescence and re-initiation in naturally fractured rocks. GRC Trans. 28, 285
288. El Rabba, W. 1989. Experimental Study of Hydraulic Fracture Initiated from Horizontal Wells. SPE Annual Technical Conference and Exhibition, San Anatonio, TX, USA. Fehler, M., 1990. Identifying the plane of slip for a fault plane solution from clustering of locations of nearby earthquakes. Geophys. Res. Lett. 17, 969
972. Ge, J., Ghassemi, A., 2008. Analysis of failure potential around a hydraulic fracture in jointed rock. Proc. 42nd U.S. Rock Mechanics Symp. San Francisco, CA. Geertsma, J., de Klerk, F., 1969. A rapid method of predicting width and extent of hydraulically induced fractures. JPT. 21, 1571
1581. Ghassemi, A., Suarez-Rivera, R., 2012. Sustaining Fracture Area and Conductivity of Gas Shale Reservoirs for Enhancing Long-Term Production and Recovery Project. RPSEA Final Report. Ghassemi, A., Rawal, A., Zhou, X., 2013. Rock failure and micro-seismicity around hydraulic fractures. J. Pet. Sci. Engrg. (108), 118
127. Available from: http://dx.doi.org/ 10.1016/j.petrol.2013.06.005. Hoek, E., Bieniawski, Z.T., 1984. Brittle fracture propagation in rock under compression. Int. J. Fracture 26, 276
294. Hope, L.C., Bratton, T., Park, Y., 2013. Completion Optimization of Unconventional Shales: A Niobrara Case Study. SPE- ATCE, Louisiana, USA, pp. 1
13. Horii, S., Nemat-Nasser, S., 1986. Brittle failure in compression: splitting, faulting and brittle-ductile transition. Philosoph. Trans. Roy. Soc. London 319 (1549), 337
374. Jacobs, T. 2014. The Shale Evolution: Zipper Fracture Takes Hold, JPT Technology Issue. Jung, R. EGS
Goodbye or Back to the Future. 2013. Presented at International Conference for Effective and Sustainable Hydraulic Fracturing. Intech, Brisbane, Australia. 20
22 May 2013.

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Kamali, A., Ghassemi, A. 2016a. On the reservoir stimulation mechanisms in fractured reservoirs. Proc. 50th US Rock Mechanics/Geomechanics Symposium held in Houston, TX. Kamali, A., Ghassemi, A. 2016b. Analysis of Natural Fracture Shear Slip and Propagation in Response to Injection. Proc. 41st Stanford Geothermal Workshop. Koshelev, V., Ghassemi, A., 2003a. Numerical modeling of stress distribution and crack trajectory near a fault or a natural fracture. Soil-Rock America Symp. Boston, MA. Koshelev, V., Ghassemi, A., 2003b. Hydraulic fracture propagation near a natural discontinuity. Proc. 28th Workshop on Geothermal Reservoir Engineering Stanford University. Stanford, California. Kumar, D., Ghassemi, A., 2016. Three-Dimensional modeling and analysis of sequential and simultaneous hydraulic fracturing of horizontal wells. J. Pet. Sci. Eng. 146, 1006
1025. Lam, K.Y., Cleary, M.P., 1984. Slippage and re-initiation of (hydraulic) fractures at frictional interfaces. Int. J. Num. Anal. Meth. Geomech. 8, 589
604. Lee, T.S., Advani, S.H., Pak, C.K., 1994. Three-dimensional hydraulic fracture simulation using fixed grid finite element algorithms. Trans. ASME. 116, 1
9. Li, Y., Chen, L., Wang, Y., 2005. Experimental research on pre-cracked marble under compression. Int. J. Solids Struct. 42 (9
10), 2505
2516. Maldonado, A., Batzle, M., Sonnenberg, S., 2011. Mechanical Properties of the Niobrara Formation. AAPG Rocky Mountain Section Meeting, Cheyenne, Wyoming, pp. 25
29. Mayerhofer, M.J., Richardson, M.F., Walker Jr., R.N., Meehan, D.N., Oehler, M.W., Browning Jr., R.R., 1997. Proppants? We Don’t Need No Proppants. SPE Annual Technical Conference and Exhibition, San Antonio, TX. 38611-MS. Ouyang, S., Carey, G.F., Yew, C.H., 1997. An adaptive finite element scheme for hydraulic fracturing with proppant transport. Int. J. Num. Methods Fluids. 24, 645
670. Perkins, T.K., Kern, L.R., 1961. Widths of hydraulic fractures. JPT. 13, 937
949. Petit, J., Barquins, M., 1988. Can natural faults propagate under Mode II conditions? Tectonics 7 (6), 1243
1256. Rao, Q., Sun, Z., Stephansson, Z., et al., 2003. Shear Fracture (Mode II) of Brittle Rock. Int. J. Rock Mech. Min. Sci. 40 (3), 355
375. Rafiee, M., Soliman, M.Y., Pirayesh, E., 2012. Hydraulic Fracturing Design and Optimization: A Modification to Zipper Frac. SPE 159786, presented at the SPE Eastern Regional Meeting. Kentucky, USA. Rodrigues, V.F., Neumann, L.F. Torres, D.S., 2007. Guimares de Carvalho. Horizontal Well Completion and Stimulation Techniques, SPE 108075, presented at the Latin American & Caribbean Petroleum Engineering Conference, Buenos Aires. Roff, A., Phillips, W.S., Brown, D.W., 1996. Joint structures determined by clustering microearthquakes using waveform amplitude ratios. Int. J. Rock Mech. Mining Sci. Geomech. Abs. 33 (6), 627
639. Rungamornrat, J., Wheeler, M.F., Mear, M.E., 2005. A numerical technique for simulating nonplanar evolution of hydraulic fractures. SPE Annual Technical Conference and Exhibition, Dallas, Texas, USA, pp. 1
9. Rutledge, J.T., Phillips, W.S., Schuessler, B.K., 1998. Reservoir Characterization Using Oil-Production-Induced Microseismicity, Clinton County, Kentucky. Tectonophysics. 289, 129
152. Safari, R., & Ghassemi, A., 2014. 3D Coupled Poroelastic Analysis of Multiple Hydraulic Fractures. Proc. 48th U.S. Rock Mechanics/Geomechanics Symposium, 1
4 June, Minneapolis, Minnesota Scavia, C., 1995. A method for the study of crack propagation in rock structures. Geotehcnique 45 (3), 447
463.

Application of rock failure simulation in design optimization of the hydraulic fracturing

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Sesetty, V.K., Ghassemi, A., 2015. Modeling and analysis of sequential and simultaneous hydraulic fracturing in single and multi-lateral horizontal wells. Int. J. Pet. Sci. Eng. 132, 65
76. Sesetty, V., Ghassemi, A., 2016. Numerical Modeling of Hydraulic Fracture Propagation from Horizontal Wells in Anisotropic Shale, Proc. 50th U.S. Rock Mechanics/ Geomechanics Symposium, Houston, TX, USA. Shen, B., Stephansson, O., 1994. Modification of the G-criterion for crack propagation under subjected to compression. Eng. Fract. Mech. 47 (2), 177
189. Soliman, M.Y., Loyd, E., David, A., 2008. Geomechanics aspects of multiple fracturing of horizontal and vertical wells. SPE Drilling Completion 23 (3), 217
228. Tarasovs, S., Ghassemi, A., 2014. Self-similarity and scaling of thermal shock fractures. Phys. Rev. E. 90 (1), 012403-1-6. Thiercelin M., Rogiers, J.C., Boone, T.J., Ingraffea, A.R., 1987. An investigation of the material parameters that govern the behavior of fractures approaching rock interfaces. Proc. of Sixth Int. Congress on Rock Mech. 1: 263
269. Montreal: Canada. Vandamme, L., Curran, J.H., 1989. A three-dimensional hydraulic fracturing simulator. Int. J. Num. Methods Eng. 28, 909
927. Vesarhelyi, B., Bobet, A., 2000. Modeling of crack initiation, propagation and coalescence in uniaxial compression. Eng. Fracture Mech. 47 (2), 177
189. Warpinski, N.R., Branagan, P.T., 1989. Altered-Stress Fracturing. JPT. Waters, G., Dean, B., Downie, R., Kerrihard, K., Austbo, L., McPherson. B., 2009. Simultaneous hydraulic fracturing of adjacent horizontal wells in the Woodford shale. SPE 119635, SPE Hydraulic Fracturing Technology Conference and Exhibition, Woodlands, Texas, USA. Wiles, T.D., Curran, J.H., 1982. A general 3D displacement discontinuity method. Proc. 4th Int. Conference for Num. Method in Geomech. 103
111. Wong, S.W., Geilikman, M., Xu, G., 2013. Interaction of multiple hydraulic fractures in horizontal wells. SPE Middle East Unconventional Gas conference and Exhibition, Muscat, Oman 1
10. Wu, H., Chudnovsky, A., Dudley, J.W. et al. 2004. A Map of Fracture Behavior in the Vicinity of an Interface. Paper presented at the Gulf Rocks 2004, the 6th North America Rock Mechanics Symposium (NARMS), Houston, Texas. ARMA 04
620. Yew, C.H., 1997. Mechanics of Hydraulic Fracturing. Gulf Publishing, Houston, TX, ISBN: 978-0-88415-474. Zhang, X., Jeffrey, R.G., 2006. The role of friction and secondary flaws on deflection and re-initiation of hydraulic fractures at orthogonal pre-existing fractures. Geophys. J. Int. 166, 1454
1465.

Anisotropic poroplasticity in saturated porous media, effect of confining pressure, and elevated temperature

2

Jianfu Shao, Shouyi Xie, Gilles Duveau, Moustafa Masri and Liang Chen University of Lille 1, Villeneuve d’Ascq, France

2.1

Introduction

Most rocks are porous materials which are generally saturated by different fluid phases. Mechanical behaviors of porous rocks are affected by variations of fluid pressures. The flow kinetics of each fluid phase is influenced by rock deformation. There is an inherent poromechanical coupling to be taken into account. In the case of an elastic behavior, the pioneer work by Biot (1941, 1973) founded the theory of poroelasticity. This theory has been reformulated and completed with the thermodynamics framework of open systems by (Coussy, 1995, 2004). The poroelastic theory is now widely applied to engineering applications and extended to anisotropic and damaged materials (Cheng, 1997; Shao, 1997; Thompson and Willis, 1991; Lydzba and Shao, 2000). Concerning plastic behaviors of porous materials, efforts have also been undertaken during the last few decades. In order to formulate plastic models for porous materials in saturated and partially saturated conditions, it is necessary to account for effects of interstitial fluid pressures on plastic yield and failure criterion. The basic idea is to extend plastic models developed for dry materials to saturated or partially saturated ones. Various approaches can be considered. Limiting our interest to plastic deformation of saturated materials, one attractive approach is the generalization of elastic effective stress concept to plastic modeling. This is the so-called stress equivalence principle. The plastic deformation of porous materials can be described by using the same plastic models as those for dry materials provided replacing the nominal stress tensor by an effective one. This concept has been used for saturated and partially saturated materials (Coussy, 1995; Coussy, 2004; Kerbouche et al., 1995; Hoxha et al., 2007; Muraleetharan et al., 2009; Chen et al., 2009). However, the validity of the effective stress concept for plastic deformation is neither theoretically proven nor experimentally verified. Some micromechanical analyses have shown that the plastic yielding, and failure criteria can be formulated as functions of effective stresses only for some particular cases of materials microstructure and loading paths (De Buhan and Dormieux, 1996, 1999; Lydzba and Shao, 2002). Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00002-6 © 2017 Elsevier Ltd. All rights reserved.

28

Porous Rock Failure Mechanics

On the other hand, many rocks are anisotropic materials. Their mechanical behavior depends on the loading orientation with respect to their material fabric. Such a material anisotropy can also affect the poromechanical coupling and the play an important role in poromechanical analysis of engineering problems. Finally, the plastic deformation and failure strength of rock may also influenced by temperature elevation, which is frequently encountered in different applications. It is, therefore, important to characterized effects of temperature change on mechanical behaviors of anisotropic rocks. In this chapter, the theoretical background of poroplasticity of anisotropic saturated materials is first recalled. The concept of effective stress is revisited and discussed. Some experimental data are presented to study the validity of such a concept on a typical porous rock. Among different methods available for modeling plastic behavior of anisotropic rocks, the approach based on the material fabric tensor is presented and applied to a sedimentary rock. Finally, some typical experimental results are presented to show the effects of temperature variation on the elastic, plastic, and failure properties of a typical shale rock.

2.2

General framework of poroplastic modeling

We recall here the general framework of poroplasticity theory for saturated porous materials with the assumption of small evolutions and isothermal conditions. Consider a representative volume element subjected to a macroscopic stress σ and an interstitial pore pressure p. Denote ε and m the total strain and fluid mass exchange per unit volume. Each of them is decomposed into an elastic part εe ,me , and a plastic part εp , mp : ε 5 εe 1 εp ;

m 5 me 1 mp

(2.1)

It is also convenient to introduce the plastic change of porosity as follows: φp 5

mp ρ0f

(2.2)

where ρ0f denotes the referential volumetric mass of interstitial fluid. Denote now Vp as a set of internal variables related to the plastic dissipation process. By assuming a linear elastic behavior, the free energy of saturated porous medium is expressed by (Coussy, 1995, 2004): ! !2 1 m 1 m p p Ψ 5 ðε 2 εp Þ:ℂ:ðε 2 εp Þ 1 Ψ p ðVp Þ 2 M 0 2 φ B:ðε 2 εp Þ 1 M 0 2φ 2 2 ρf ρf (2.3)

Anisotropic poroplasticity

29

The fourth rank tensor ℂ defines the elastic stiffness under undrained conditions, M denotes the Biot’s modulus, and B is a second-order tensor of Biot’s coefficients. The first term of the right hand side represents the energy of dry porous skel strain  eton without interstitial fluid. The function Ψp Vp defines the locked energy for plastic process. The last two terms are introduced to describe the poroelastic coupling in a saturated porous medium. The elastoplastic constitutive equations of porous medium are deduced by the standard derivation of the free energy function: m σ 5 ℂ:ðε 2 ε Þ 2 BM 2 φp ρf

!

p

" p5M

(2.4)

# ! m 2 φp 2 B:ðε 2 εp Þ ρ0f

(2.5)

Substituting (2.5) for (2.4), the stressstrain relations can be also expressed by: σ 5 ℂb :ðε 2 εp Þ 2 Bp

(2.6)

The fourth-order tensor ℂb denotes the elastic stiffness in drained conditions. It is easy to obtain: ℂb 5 ℂ 2 MB  B

(2.7)

The intrinsic dissipation should verify the following positive condition: @Ψ p k σ:_εp 1 pφ_ 1 Akp V_ p $ 0; Akp 5 2 k @Vp

(2.8)

The set of variables Akp denotes the conjugate thermodynamics forces associated with the internal variables Vpk . The objective of poroplastic modeling is to determine p k the incremental or rate values of the state variables (_εp ; φ_ ; V_ p ). In the case of time independent dissipation, it consists in defining the plastic yield function that describes the elastic convex boundary. Generally, this is a scalar function of the conjugated force variables:   f σ; p; Akp 5 0

(2.9)

For most rock-like materials, the assumption of the generalized standard material is generally not verified. The plastic flow rule should be defined with the help of a nonassociated plastic potential Qðσ; p; Vpk Þ 5 0 (see Fig. 2.1) so that: ε_ p 5 λ_

  @Q σ; p; Vpk @σ

;

p φ_ 5 λ_

  @Q σ; p; Vpk @p

;

  _ σ; p; V k γ_ p 5 λh p

(2.10)

30

Porous Rock Failure Mechanics

σkl k

G(σ,p,Ap ) = 0 k

f(σ,p,A p ) = 0 σij

p

Figure 2.1 Illustration of determination of yield function and of the plastic potential in the stress and interstitial pressure space.

The plastic multiplier λ_ . 0 is determined with the plastic consistency condition as in the case of the classic plastic theory. The function h defines a specific plastic hardening law for the porous medium.

2.2.1 Effective stress concept in poroplasticity As shown above, the plastic yield function and potential are both functions of two independent forces, the total stress and interstitial pressure. The determination of such functions requires laboratory tests involving complex loading paths in the space of stress and interstitial pressure. Due to the technical limitation, only a small number of tests can be practically performed. Therefore, it is desired to establish a simplified approach. The basic idea is to extend a plastic model for dry materials to saturated ones by introducing a concept of effective stress. The principle of stress equivalence is then introduced and stipulates that the plastic functions for a saturated porous medium can be expressed in the same forms as those for the dry skeleton material by replacing the total stress tensor by a suitable effective stress tensor (Coussy, 1995, 2004). This principle is implicitly used in soil mechanics through the classical Terzaghi’s effective stress tensor. However, in cohesive rock-like materials, the validity of such a principle is so far neither theoretically nor experimentally fully proven. Some micromechanical studies have been undertaken and have provided some useful clarification. De Buhan and Dormieux (1996, 1999) have shown that the failure criterion of a fluid saturated porous medium could be expressed in terms of the concept of effective stress and the expression of effective stress tensor depended on the local failure condition of the solid matrix. Lydzba and Shao (2000) have obtained similar results for the initial plastic yield criterion of fluid saturated porous medium. More general cases have been studied by Lydzba and Shao (2002) regarding the validity of the effective stress concept in evolving plastic hardening domain. The form of the effective stress tensor has been examined for two particular classes of skeleton materials. In the case of a microhomogeneous

Anisotropic poroplasticity

31

and microisotropic skeleton material verifying the von-Mises criterion, the stress equivalence principle is valid for the whole plastic domain and the Terzaghi’s effective stress is the thermodynamic force associated with the plastic strain and can be applied to any nonassociated flow rules. However, if the material skeleton obeys a pressure-sensitive MohrCoulomb type criterion, the validity of the stress equivalence principle cannot be proven for the general case. The existence of an effective stress tensor can be demonstrated only for the case of a constant interstitial pressure during the entire loading path. This effective stress depends not only on the fluid pressure, but also the frictional angle and cohesion of skeleton material. Therefore, the effective stress concept for poroplastic modeling of fluid saturated materials is theoretically validated only for some particular cases. In the framework of a phenomenological approach, the effective stress concept in poroplasticity has been also discussed, based on some kinematical assumptions. For instance, it is assumed that the rate of plastic porosity is proportional to the rate of plastic volumetric strain (Biot, 1973; Coussy, 1995, 2004; Kerbouche et al., 1995): p φ_ 5 β:_εp ;

trβA½0; 3

(2.11)

The components of the second-order tensor β define the kinematic relationship the plastic volumetric strain rate and plastic porosity variation rate. The positiveness of the intrinsic dissipation can be now written as ðσ 1 βpÞ:_εp 1 Akp V_ p $ 0 k

(2.12)

Therefore, the quantity ðσ 1 βpÞ appears as the thermodynamic force associated with the plastic strain tensor. In this sense, this quantity can be seen as the effective stress tensor for the plastic flow rule. As a consequence, the plastic potential can be written in terms of this effective stress: Gðσ; p; Akp Þ  Gðσpl ; Akp Þ;

σpl 5 σ 1 βp

(2.13)

In general, the components of β are not equal those of the Biot’s coefficient tensor B, which was defined in the poroelastic law. If an associated flow rule is adopted, the yield function can also be expressed with the plastic effective stress:     f σ; p; Akp  f σpl ; Akp # 0

(2.14)

However, for most rock-like materials, a nonassociated flow rule is generally needed. The kinematic assumption does not imply that the yield function can be formulated in terms of the effective stress. Therefore, further theoretical and experimental investigations are still needed. In this chapter, we present some experimental data performed in view of checking the validity of effective stress concept in a typical porous rock. However, the isotropic assumption is adopted for this material.

32

2.3

Porous Rock Failure Mechanics

Experimental investigation on a typical porous rock

The typical porous rock considered here is a white chalk mainly composed of calcite (.98%) and quartz (,2%), with an average porosity of 43% (Xie and Shao, 2006, 2012). The first test is a hydrostatic compression test with a number of stresspressure cycles. Each cycle is composed of three steps as illustrated in Fig. 2.2 (Yamada et al., 1981). The point 1 denotes a state of stress and pressure on the current yield surface. During the steps 12, the hydrostatic stress (confining pressure) is increased under an undrained condition. The generated variations of pore pressure and volumetric strain are measured. Both elastic and plastic strains are produced during this step. In the steps 23, the confining pressure is unloaded to its initial value at the point 1 and one reaches the point 3. Due to the plastic strain, the pore pressure at the point 3 is generally not equal to its initial value at the point 1. Therefore, in the steps 34, some quantity of fluid is either injected or expulsed in order to bring the pore pressure to its value at the point 1. In this way, a complete cycle of poromechanical loading and unloading is archived. The expulsed or injected fluid quantity during the step 34 represents the irreversible fluid mass change which can be used to calculate the plastic porosity change, as defined by the relation (2.2). On the other hand, the plastic volumetric strain is also measured by strain gages. The plastic effective stress coefficient (β 5 βδ for an isotropic material) corresponds to the ratio between the plastic porosity and volumetric strain rates as defined in (2.11). In Fig. 2.3, the evolutions of plastic volumetric strain and plastic porosity are presented. One can see that the two quantities are very close each other. This seems to suggest that the plastic effective stress for the porous chalk is close to 1 (β 5 1). As a consequence, the Terzaghi’s stress can be used as the effective stress for this particular rock under hydrostatic compression. For the same objective as that of the hydrostatic test, a triaxial compression test including a series of stresspressure cycles. The principle of each cycle is presented in Fig. 2.4. From the point 1 which is on the current yield surface, an increment of axial stress Δσ1 is applied by keeping the lateral stress and pore pressure Pc 2

1

4

3 Pi

0

0

Figure 2.2 Illustration of loading steps in hydrostatic poromechanical tests (Yamada et al., 1981).

Anisotropic poroplasticity

33

60

Hydrostatic stress (MPa)

50 40 30

Red circle-plastic porosity

20 10

Plastic volumetric strain (%) 0 0

5

10

15

Figure 2.3 Comparison between plastic volumetric strain (line) and plastic porosity during hydrostatic poromechanical loading (Xie and Shao, 2012).

σ'1,σ'3 5 2 4

3

1 Pi

Figure 2.4 Illustration of a poroplastic loading—unloading cycle in triaxial compression test (Yamada et al., 1981).

constant until the point 2. Then, the applied axial stress increment is entirely unloaded and one obtains the point 3. From this point, the axial stress is kept constant, while the pore pressure and lateral stress are decreased of the same (absolute) value as that of Δσ1 and the point 4 is reached. Note that the plastic effective stress state is identical at the points 2 and 4. The objective here is to check whether these two points are located on the same yield surface. In Fig. 2.5, the experimental results are presented for the test with 15 MPa initial confining pressure and 5 MPa initial interstitial pressure. On can see that the rock exhibits plastic strains and hardening during the loading step [(2.1) and (2.2)]. The response of rock is quasi linear and elastic during the subsequent steps until the point 4. Plastic strains occur again beyond the point 4. Therefore, the points 2 and 4 are located on the same yield surface. Further, the axial and lateral plastic strains are very close between these two loading points. As we have the same plastic effective stresses at these two points, it is possible to say that the plastic flow of rock is controlled by the effective stress defined above, at least for this specific loading path.

34

Porous Rock Failure Mechanics

18

σ1–σ3

σ1–σ3 (MPa)

16 14

Pc

12 10 8 6 Pi

4 2

ε3(%)

ε1(%)

0

–4

0

4

8

12

Figure 2.5 Stress strain curves obtained from test with 15 MPa initial confining pressure and 5 MPa initial interstitial pressure—deviatoric stress, confining pressure and interstitial pressure versus axial strain 1 Lateral strain a` superposer! (Xie and Shao, 2012).

4

Deviatoric

3 2 1 0 0

2

4

6

8

Effective mean stress (MPa)

Figure 2.6 Yield stresses obtained from triaxial compression tests performed respectively without (empty symbols) and with pore pressure (filled symbols) (Xie and Shao, 2012).

As another example to check the validity of effective stress for the plastic yield criterion, a series of drained triaxial compression tests are performed, respectively, without and with a constant interstitial pressure. The principle is to choose particular values of confining pressure and interstitial pressure so that their difference is the same as the confining pressure used in a test without interstitial pressure. Then, we compare the yield stress and failure strength between these two tests in order to study the effects of interstitial pressure. In Fig. 2.6, one can see that the yield stresses obtained from different tests are very close between the two series of tests. This means that the plastic yield of the porous rock studied here is essentially controlled by the effective confining pressure. This result confirms in a different way the validity of effective stress for plastic modeling of the porous chalk, at least under drained triaxial compression condition.

Anisotropic poroplasticity

2.4

35

Anisotropic plastic behavior of rocks

Many rocks exhibit an anisotropic mechanical behavior. Consider here the class of sedimentary rocks, such as shale, siltstone, and claystone. The study of the mechanical behavior of these rocks is of particular interest to the oil exploration industry, to the geological disposal of radioactive waste as well as to civil and mining engineering. These rocks exhibit a strong inherent anisotropy, characterized by a directional dependence of deformation characteristics. The anisotropy is inherently related to the microstructure, in particular the existence of bedding planes. Over the last few decades, an extensive research effort has been devoted to study the mechanical behavior of anisotropic rocks. Comprehensive references on this topic can be found in a number of review papers, for instance Amadei (1983), Kwasniewski (1993), and Ramamurthy (1993). Extensive experimental studies have been undertaken on various anisotropic rocks. The results from uniaxial and triaxial compression tests generally indicate that the maximum axial compressive strength is associated with the configurations in which the bedding planes are either parallel or perpendicular to the loading direction. At the same time, the minimum strength is typically associated with the failure along the weakness plane, which corresponds to sample orientations within the range 30 to 60 . In Fig. 2.7, one can see the axial compressive strength of the Tournemire shale for different confining pressures (Niandou et al., 1997). In parallel with experimental studies, extensive research has been carried out on formulation of appropriate general failure criteria. An extensive review on this topic, examining different approaches, is provided in Duveau et al. (1998). In general, relatively little work has been done on the description of progressive failure in

) MPa

120 50 MPa 100

40 MPa 20 MPa

Failure (Peak) stress (

80 60

5 MPa 1 MPa

40 20 0 0

10

20

30

40

50

60

70

80

90

100

Orientation of the bedding plane

Figure 2.7 Variation of failure stress with loading orientation under different confining pressures of Tournemire shale (Niandou et al., 1997).

36

Porous Rock Failure Mechanics

this class of materials, which stems primarily from difficulties associated with the formulation of the problem. The rigorous approach, based on general representation theorems for tensorial functions (Boehler and Sawczuk, 1970, 1977), is very complex and has never been applied to any practical problem. On the other hand, other studies have been devoted to the development of approaches which retain the mathematical rigor and, at the same time, are pragmatic, that is, simple enough to be applied to solve some practical engineering problems. Pietruszczak and Mroz (2001) have proposed a formulation incorporating a scalar anisotropy parameter which is expressed in terms of mixed invariants of the stress and structureorientation tensors. This approach has been successively applied to describe plastic deformation of the Tournemire shale Pietruszczak et al. (2002). In this chapter, this approach is presented as a representative example. Consider first an anisotropic material with an orthotropic symmetry. In order to represent the material anisotropy, a second-order microstructure tensor (a) is introduced with its three principal values (a1 ; a2 ; a3 ). Define now the principal triad of ! ! ! the microstructure tensor (a) by (S1 ; S2 ; S3 ), as shown in Fig. 2.8. Denote σij the components of Cauchy stress tensor in the principal frame and specify the traction moduli on the planes normal to the principal axes ! (Si ; i 5 1; 2; 3):  1=2  1=2  1=2 L1 5 σ211 1σ212 1σ213 ; L2 5 σ221 1σ222 1σ223 ; L3 5 σ231 1σ232 1σ233 (2.15) The normalized load vector is then defined as follows: li 5

Li ðLk Lk Þ1=2

;

ð2Þ ð3Þ Li 5 L1 Sð1Þ i 1 L2 Si 1 L3 Si

(2.16)

Let now introduce a scalar anisotropy parameter (η), which represents the projection ! of the microstructure tensor (a) on the current loading direction (l ) defined in (2.16): η 5 aij li lj 5

trðaσ2 Þ trðσ2 Þ

(2.17)

S1

l

S3

S2 !

Figure 2.8 Illustration of structure frame and loading vector l .

Anisotropic poroplasticity

37

In practical geotechnical engineering problems, rocks are generally subjected to in situ initial stresses (σ0ij ). Plastic deformation and failure process are generated by stress variations with respect to the initial state. It is then convenient to define the loading orientation with the stress variations. The traction moduli defined in (2.15) and used in (2.16) is modified as follows:  1=2 L1 5 σ~ 211 1 σ~ 212 1 σ~ 213 ;  1=2 L3 5 σ~ 231 1 σ~ 232 1 σ~ 233 ;

 1=2 L2 5 σ~ 221 1 σ~ 222 1 σ~ 223 ; σ~ ij 5 σij 2 σ0ij

(2.18)

As for any symmetric second-order tensor, the microstructure tensor can be decomposed into a spherical part and deviatoric part: aij 5 η^ ðAij 1 δij Þ;

η^ 5

1 akk 3

(2.19)

traceless symmetric tensor which It is obvious that Aij are components of a ! describes the bias in the spatial distribution of ηðl Þ with respect to the mean value η^ . Using such decomposition, the scalar anisotropy parameter given in (2.17) can be expressed in the following form: η 5 aij li lj 5 η^ ð1 1 Aij li lj Þ

(2.20)

In order to better describe the loading orientation dependency, the above anisotropy parameter can be extended to a more general expression by invoking higher order microstructure tensors:   η 5 η^ 1 1 Aij li lj 1 Aijkl li lj lk ll 1 Aijklmn li lj lk ll lm ln 1 ?

(2.21)

As proposed by Pietruszczak et al. (2002), a specific case of this representation of microstructure is to define the high-order tensors via dyadic products of A, that is, Aijkl 5 b1 Aij Akl ; Aijklmn 5 b2 Aij Akl Amn : h i  2  3  4 η 5 η^ 1 1 Aij li lj 1 b1 Aij li lj 1 b2 Aij li lj 1 b3 Aij li lj 1 ?

(2.22)

With the notion of anisotropy parameter, it is now possible to determine the plastic yield function and plastic potential of anisotropic materials in general loading conditions. As an example, consider the following nonlinear yield function for sedimentary rocks (Chen et al. 2010):   p m f 5 q 2 βðγ p Þfc ðηÞ Cs 1 50 fc ðηÞ

(2.23)

38

Porous Rock Failure Mechanics

p5

2 σkk ; 3

q5

pffiffiffiffiffiffiffi 3J2 ;

J2 5

sij sij ; 2

sij 5 σij 2

σ  kk δij 3

(2.24)

The parameters Cs and m define the material cohesion and frictional angle associated with the yield surface. The function βðγ p Þ defines the plastic hardening law with the generalized plastic strain γ p . The coefficient fc presents the uniaxial compression strength of material which is used here as a normalization parameter. In order to introduce the influence of loading orientation on the plastic yield criterion of anisotropic materials, it is assumed that the uniaxial compression strength is a function of the anisotropy parameter (η). Based on the representation given in (2.22), the distribution function of the uniaxial compression strength (fc ) is given by: h i  2  3  4 fc 5 f^c 1 1 Aij li lj 1 d1 Aij li lj 1 d2 Aij li lj 1 d3 Aij li lj 1 ?

(2.25)

One can see that the parameter (f^c ) represents the average value of uniaxial compression strength with loading orientation. For the case of the Tournemire shale, due to the existence of parallel bedding planes, a transversely isotropic structure is generally assumed. Thus, one has A2 5 A3 5 2 0:5A1 . Now, from the best fitting of experimental data shown in Fig. 2.7, the following values of parameters are obtained (Chen et al., 2010): A1 5 0:017;

f^ 5 22 MPa;

d1 5 515;

d2 5 61735;

d3 5 2139820

The comparison between experimental data and theoretical prediction is presented in Fig. 2.9. In order to show effects of material anisotropy on poroplastic coupling, an example of numerical analysis of a coupled hydromechanical problem is now presented. It is the excavation of a circular horizontal gallery in a saturated porous rock formation. The gallery is excavated in the direction of minor principal stress. The crosssection of the gallery and geometrical domain is presented in Fig. 2.10. Note that the bedding planes of the rock are parallel to the horizontal direction. The initial in situ stresses and pore pressures are σv 5 2 12:7 MPa;

σh 5 2 12:4 MPa;

σH 5 2 16:12 MPa;

pw 5 4:7 MPa

For the sake of simplicity, the effect of body force is neglected. The excavation process is classically described by the progressive diminution of radial stress and pore pressure on the gallery wall from their initial values to atmospheric pressure. A series of calculations have been performed respectively assuming an isotropic or anisotropic behavior for the rock. Only some selected results are presented here in order to show the effects of rock anisotropy. In Fig. 2.11, we show the distribution of generalized plastic strain respectively along the horizontal and vertical axes of the cross section with the comparison between an isotropic and anisotropic model. As the effective horizontal stress is

Anisotropic poroplasticity

S1

39

y

S2 x

α

fc(MPa)

50

40

30

20 Loading orientation (degree) 10 0

15

30

45

60

75

90

Figure 2.9 Variation of uniaxial compressive strength with loading orientation (α) for Tournemire shale (Chen et al., 2010).

50 m

r = 2.6 m

σx = –16.12 MPa, pi = 4.7 MPa

Galerie circulaire

50 m

Dx = 0

σy = –12.7 MPa, pi = 4.7 MPa

y x

Dy = 0

Figure 2.10 Geometrical domain and boundary conditions.

higher than that in the vertical direction, the most important plastic zone is situated around the gallery roof. The plastic strain at the roof is then higher than that at the foot. This difference of plastic strain is due to the anisotropy of in situ initial stresses. However, when an anisotropic plastic model is used as that described above, the difference of plastic strain between two directions becomes much smaller than

40

Porous Rock Failure Mechanics

γp

4,00E-03 3,50E-03 3,00E-03 2,50E-03

Ref Plas aniso

2,00E-03 1,50E-03 1,00E-03 5,00E-04

x (m)

0,00E+00 2

3

4

5

6

7

8

9

10

8

9

10

γp

8,00E-03

6,00E-03 Ref Plas aniso

4,00E-03

2,00E-03

y (m)

0,00E+00 2

3

4

5

6

7

Figure 2.11 Distribution of generalized plastic strain along horizontal and vertical axes after excavation, comparison between isotropic and anisotropic calculations.

that obtained by the isotropic model. Therefore, the plastic deformation process is affected by the material anisotropy. In Fig. 2.12, the distribution of fluid pressure is presented. Again, one can see that the results are influenced by the material anisotropy. For the isotropic model, there is an over-pressure in the roof zone due to compressive plastic deformation. For the anisotropic model, as the plastic deformation is less important in that zone, there is no over-pressure observed. However, as the elastic modulus is smaller in the perpendicular direction than that in the parallel direction to bedding planes, the elastic compressive deformation is more important in the foot zone than in the roof zone. One obtains an over-pressure around the foot point for the anisotropic model.

Anisotropic poroplasticity

7,50E+06

41

pl (Pa)

6,00E+06

4,50E+06 Ref

3,00E+06

Plas aniso 1,50E+06 y (m)

0,00E+00 0

10

20

30

40

50

60

–1,50E+06 6,00E+06 p (Pa) l 5,00E+06 4,00E+06 Ref Plas aniso

3,00E+06 2,00E+06 1,00E+06 x (m)

0,00E+00 0

10

20

30

40

50

60

Figure 2.12 Distribution of liquid pressure along horizontal and vertical axes after excavation, comparison between isotropic and anisotropic calculations.

Finally, the radial displacements along two directions are given in Fig. 2.13, and they are also dependent on the rock anisotropy.

2.5

Effects of temperature on anisotropic rocks

In this section, we present some experimental results on the effects of temperature on an anisotropic sedimentary rock, the Tournemire shale. A series of hydrostatic and triaxial compression tests have been performed under different temperatures until 250 C (Masri et al., 2014). In Fig. 2.14, axial and radial strains are presented

42

Porous Rock Failure Mechanics

0,00E+00 0

10

20

30

40

50 x (m) 60

–5,00E-03

–1,00E-02 Ref Plas aniso

–1,50E-02

–2,00E-02

ux (m) –2,50E-02

0,00E+00

0

10

20

30

40

50

y (m)

60

–3,00E-03

–6,00E-03

–9,00E-03

Ref Plas aniso –1,20E-02

–1,50E-02

–1,80E-02

uy (m)

Figure 2.13 Distribution of radial displacement along horizontal and vertical axes after excavation, comparison between isotropic and anisotropic calculations.

as function of hydrostatic stress. First, these results confirm the anisotropic deformation properties of the shale. The strain in the perpendicular direction (axial strain) is much higher than that in the parallel direction (radial strain). Further, the deformation of rock becomes higher when the temperature increases. Therefore, the temperature enhances the deformability of the shale. In Figs. 2.15 and 2.16, we show the typical stressstrain curves of the studied shale in triaxial compression tests with a 5-MPa confining pressure and under different temperatures, in two loading direction. It is clear that the mechanical

Anisotropic poroplasticity

43

20

15 axial (20 °C) radial (20 °C) axial (100 °C) radial (100 °C) axial (150 °C) radial (150 °C) axial (200 °C) radial (200 °C) axial (250 °C) radial (250 °C)

10

5

Strains

0 0

0.002

0.004

0.006

0.008

0.01

Figure 2.14 Axial (perpendicular) and radial (parallel) strains under different temperatures in hydrostatic compression test on Tournemire shale. Deviatoric stress (MPa)

70

Pc=5 MPa

60

σ1 σ3

50

σ3

40 30 T=20 °C 20

T=100 °C T=150 °C

10

T=200 °C

Radial strain

T=250 °C

Axial strain 0

–0.01

–0.005

0

0.005

0.01

0.015

Figure 2.15 Stressstrain curves during triaxial compression tests with θ 5 90 and 5 MPa confining pressure and with different values of temperature.

response of the shale is strongly affected by the temperature. The elastic modulus and peak strength both decrease with the temperature increase. Further, it seems that the shale behavior becomes more ductile under higher temperature. Finally, in Fig. 2.17, the peak stresses obtained in triaxial compression tests in the perpendicular direction are drawn. It is clear that the mechanical strength of the shale

44

Porous Rock Failure Mechanics

Deviatoric stress (MPa) 60

Pc = 5 MPa

σ1

50

σ3

40

σ3

T = 20 °C T = 100 °C T = 150 °C T = 200 °C T = 250 °C

30

20

10 Radial strain

Axial strain 0

–0.01

–0.005

0

0.005

0.01

0.015

Figure 2.16 Stressstrain curves during triaxial compression tests with θ 5 0 and 5 MPa confining pressure and with different values of temperature.

100

Deviatoric stress (MPa)

90 80 70

σ1

60

σ3

σ3

50 40

T = 20 °C

30

T = 100 °C

20

T = 150 °C T = 200 °C

10

T = 250 °C

Mean stress (MPa)

0 0

10

20

30

40

50

60

Figure 2.17 Failure surface in p 2 q plane obtained for loading orientation θ 5 90 and at different values of temperature.

Anisotropic poroplasticity

45

significantly depends on the temperature. Both the frictional angle and cohesion decrease with the temperature increase.

2.6

Conclusions

In this chapter, we have presented a short review of poroplastic theory for saturated porous materials and discussed the effects of material anisotropy and temperature on the mechanical behavior of a class of sedimentary rocks. The effective stress concept is widely used in numerical modeling of engineering problems. However, its validity is not completely demonstrated neither form theoretical point of view nor from experimental evidences. Further investigations are needed from both micromechanical and phenomenological studies. The rock structural anisotropy may play a crucial role in hydromechanical modeling. Various approaches are available for the description of plasticity a failure in anisotropic rocks. However, relevant experimental data are still needed and essential for a pertinent description of anisotropy effects. The temperature variation can significantly affect the mechanical behavior of rocks, in particular sedimentary rocks. In general, the elastic modulus and failure strength decrease with the temperature increase. Further, the material anisotropy effects may be also influenced by the temperature change. Under a high temperature, the material anisotropy can be attenuated.

References Amadei, B., 1983. Rock Anisotropy and the Theory of Stress Measurements. Springer, Berlin. Biot, M.A., 1941. General theory of three dimensional consolidation. J. Appl. Phys. 12, 155164. Biot, M.A., 1973. Non-linear and semi-linear rheology of porous solids. J. Geophy. Res. 78, 49244937. Boehler, J.P., Sawczuk, A., 1970. Equilibre limite des sols anisotropes. J. de Mecanique. 3, 533. Boehler, J.P., Sawczuk, A., 1977. On yielding of oriented solids. Acta Mech. 27, 185206. Chen, L., Rougelot, T., Chen, D., Shao, J.F., 2009. Poroplastic damage modeling of unsaturated cement-based materials. Mech. Res. Commun. 36, 906915. Chen, L., Shao, J.F., Huang, H.W., 2010. Coupled elastoplastic damage modeling of anisotropic rocks. Comput. Geotech. 37, 187194. Cheng, A.H.D., 1997. Material coefficients of anisotropic poroelasticity. Int. J. Rock Mech. Min. Sci. 34 (2), 199205. Coussy, O., 1995. Mechanics of Porous Continua. John Wiley & Sons, UK. Coussy, O., 2004. Poromechanics. John Wiley & Sons, UK. De Buhan, P., Dormieux, L., 1996. On the validity of the effective stress concept for assessing the strength of saturated porous materials: a homogenization approach. J. Mech. Phys. Solids. 44, 16491667. De Buhan, P., Dormieux, L., 1999. A micromechanics-based approach to the failure of saturated porous media. Transp. Porous Media. 34, 4762.

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Duveau, G., Shao, J.F., Henry, J.P., 1998. Assessment of some failure criteria for strongly anisotropic materials. Mech. Cohesive Frictional Mater. 3, 126. Hoxha, D., Giraud, A., Homand, F., Auvray, C., 2007. Saturated and unsaturated behaviour modelling of meuse haute marne argillite. Int. J. Plast. 23, 733766. Kerbouche, R., Shao, J.F., Skoczylas, F., 1995. On the poroplastic behaviour of porous rock. Eur. J. Mech. A/Solids. 14, 577587. Kwasniewski, M.A., 1993. Mechanical behaviour of anisotropic rocks. In: Hudson, J.A. (Ed.), Comprehensive Rock Engineering, vol. 1: Fundamentals. Pergamon Press, Oxford, pp. 285312. Lydzba, D., Shao, J.F., 2000. Study of poroelasticity material coefficients as response of microstructure. Mech. Cohesive-Frictional Mater. 5, 149171. Lydzba, D., Shao, J.F., 2002. Stress equivalence principle for saturated porous media. C.R. Me´canique. 330, 297303. Masri, M., Sibai, M., Shao, J.F., Mainguy, M., 2014. Experimental investigation of the effect of temperature on the mechanical behavior of Tournemire shale. Int. J. Rock Mech. Min. Sci. 70 (2014), 185191. Muraleetharan, K.K., Liu, C., Wei, C., Kibbey, T.C.G., Chen, L., 2009. An elastoplastic framework for coupling hydraulic and mechanical behavior of unsaturated soils. Int. J. Plast. 25, 473490. Niandou, H., Shao, J.F., Henry, J.P., Fourmaintraux, D., 1997. Laboratory investigation of the mechanical behaviour of Tournemire shale. Int. J. Rock Mech. Min. Sci. 34, 316. Pietruszczak, S., Mroz, Z., 2001. On failure criteria for anisotropic cohesive-frictional materials. Int. J. Num. Analyt. Methods Geomech. 25, 509524. Pietruszczak, S., Lydzda, D., Shao, J.F., 2002. Modelling of inherent anisotropy in sedimentary rocks. Int. J. Solids Struct. 39 (3), 637648. Ramamurthy, T., 1993. Strength and modulus responses of anisotropic rocks. In: Hudson, J.A. (Ed.), Comprehensive Rock Engineering, vol. 1. Fundamentals. Pergamon Press, Oxford, pp. 319329. Shao, J.F., 1997. A numerical solution for a thermo-hydro-mechanical coupling problem with heat convection. Int. J. Rock Mech. Mining Sci. 34 (1), 163166. Yamada, S.E., Schatz, J.F., Abou Sayed, A., Jones, A.H., 1981. Elasto-plastic behavior of porous rock under undrained condition. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 18, 177179. Thompson, M., Willis, J.R., 1991. A reformulation of the equations of anisotropic poroelasticity. J. Appl. Mech. ASME. 58, 612616. Xie, S.Y., Shao, J.F., 2006. Elastoplastic deformation of a porous rock and water interaction. Int. J. Plast. 22, 21952225. Xie, S.Y., Shao, J.F., 2012. Experimental investigation and poroplastic modeling of saturated porous geomaterials. Int. J. Plast. 39 (2012), 2745.

Coupling in hydraulic fracturing simulation

3

Alexandre Lavrov SINTEF Petroleum Research, Trondheim, Norway

3.1

Introduction: fluid-driven fracture propagation in rocks

Fluid-driven fracture propagation is a multiscale, multiphysics phenomenon. Fluid injected into the fracture works to advance the fracture tip. Part of the fluid leaks through the fracture face (wall) into the surrounding porous rocks. Filter cake may build up on the fracture faces and behind the fracture tip. The fluid pressure opens up the fracture. This changes the fracture aperture and, thus, the fracture permeability. The permeability is affected not only by the average fracture aperture, but also by the tortuosity of the flow paths inside the fracture. The tortuosity is different in different rock types. It is also affected by asperities crushing caused, for example, by shear displacement of the fracture faces. If the fracturing fluid contains solid particles (proppant), the solids concentration affects the rheological properties of the slurry and, thus, influences the flow and the pressure distribution in the fracture, affecting the fracture opening. The fracture opening, in turn, affects the proppant transport as proppant particles can be detained in the narrow parts of the fracture (proppant screenout). The above brief overview shows that the most essential components of hydraulic fracturing do not work in isolation but rather are interconnected. In a numerical model, coupling must be established between parts of the model describing different aspects of physics and mechanics of hydraulic fracturing.

3.2

Coupling in reservoir geomechanics

The most basic type of coupling required for hydraulic fracturing simulation is the hydro-mechanical coupling (HM-coupling). This type of coupling is routinely used in geomechanical simulations in general when one needs to study the stress changes caused by production/injection and to investigate how these stress changes affect the fluid flow in the reservoir. In particular, in HM-coupled reservoir models, changes in the pore pressure caused by production (or injection, or both) alter the effective and total stresses in the reservoir and in the overburden. These stress Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00003-8 © 2017 Elsevier Ltd. All rights reserved.

48

Porous Rock Failure Mechanics

Figure 3.1 Hydro-mechanical coupling in coupled geomechanical simulation. Models coupled in the simulation are shown as boxes. Text in italics identifies the data passed between the models.

changes, in turn, affect the hydraulic properties of the reservoir, that is, porosity and permeability. These, again, affect the fluid flow, and so on. In reality, there is thus a two-way coupling between the reservoir flow and the geomechanics (Fig. 3.1). This two-way coupling can, in general, be expressed in the discretized form as follows (Longuemare et al., 2002; Boone and Ingraffea, 1990): KΔt u 1 LΔt P 5 F

(3.1a)

LT Δt u 1 EΔt P 5 R

(3.1b)

where u is the vector of solid displacements; P is the vector of pore pressures; F is the vector of boundary forces (nodal loads in the finite-element method); R is the vector of source terms in the flow model; K is the solid-mechanical stiffness matrix; E is the flow matrix; L is the coupling matrix between displacements and pore pressure. The discrete operator, Δt , represents the change from time step n to time step (n 1 1). The coupling term LΔt P represents the effect of pore pressure on deformations. The coupling term LT Δt u represents the effect of (volumetric) deformations on the pore pressure. In a simplest model, this two-way coupling can be replaced with one-way coupling. One-way coupling implies that there are two standalone pieces of software: a reservoir flow code (a reservoir simulator) and a geomechanical code. With one-way coupling, the reservoir simulator is run step by step, and the pressure data obtained at certain time steps are passed to the geomechanical code. These data are used in the geomechanical code to solve for stresses and displacements in a poroelastic or poroelastoplastic medium. These stresses and displacements can then be used to evaluate reservoir compaction, fault reactivation, subsidence, and borehole stability. They are not used, however, to update the reservoir properties in the reservoir flow code, such as porosity and permeability. Hence, there are no geomechanical effects on the flow model. The coupling is thus one-way, from the

Coupling in hydraulic fracturing simulation

49

flow simulator to the geomechanical simulator only. One-way coupled models are also known as decoupled models (Settari and Walters, 2001). When information is passed in both directions, that is, pressures are passed from the reservoir flow code to the geomechanical code, and stress-dependent porosities and permeabilities are passed in the opposite direction, the coupling is two-way. In this case, the coupling may be either full or partial. In a fully coupled model, the coupled Eqs. (3.1a) and (3.1b), for example, the Biot poroelasticity equations, are solved simultaneously, as one system and using one simulator (Schrefler and Scotta, 2001; Doster and Nordbotten, 2015). Such coupled models have good convergence properties but often are simplified with regard to the physical assumptions underlying the model, for example, linear elasticity only. Codes are also relatively difficult to modify and maintain. In a partially coupled model (also known as iteratively coupled model), there are two solvers that are invoked in turn (Longuemare et al., 2002). These solvers, that is, the reservoir flow model and the geomechanical model, can be implemented either as two standalone pieces of software or as two solvers within the same code. At each time step, Eq. (3.1b) is solved for pore pressure, assuming that the displacement field is given. Next, Eq. (3.1a) is solved for displacements, using the pore pressure vector obtained from Eq. (3.1b). The stress equations and the flow equations are thus solved separately, and the data are passed between the two systems in an iterative process until convergence is reached. The simulation then advances to the next time step. The accuracy of the coupled solution is determined by the number of iterations or the tolerance. If only one iteration is performed, the coupling is called explicit. If several iterations are performed until convergence is reached, the coupling is called implicit. The solution to which the implicitly coupled model converges is identical with the solution that would be obtained with a fully coupled model. Partially coupled models offer more flexible and modular software design. The geomechanical and fluid mechanical components can be maintained and upgraded independently of each other. The main advantage of a partially coupled model as compared to a fully coupled one is its modularity. This entails greater flexibility of partially coupled models, from the developer’s and user’s perspectives. During the software development cycle, incremental modifications can be made to either the reservoir flow model or the geomechanical model. A common disadvantage of partially coupled models is the use of different numerical methods and, as a result, of different grids in the reservoir flow model and in the geomechanical model. Often, the finite-volume method or the finite-difference method is employed to solve the reservoir flow problem, whereas the finite-element method is used in the geomechanical code. The former often operates on a structured grid, whereas the latter on an unstructured grid. As a consequence, it may become necessary to perform interpolation in order to exchange data between the reservoir simulator and the geomechanical code. Interpolation reduces the overall accuracy. The classification of HM-coupling methods laid out above is summarized in Fig. 3.2. In a hydraulic fracturing simulation, coupling flow and deformations is more challenging because of the strong nonlinearities present in hydraulic fracturing

50

Porous Rock Failure Mechanics

Figure 3.2 Types of hydro-mechanical coupling in reservoir geomechanics.

models. In particular, fracture permeability is a nonlinear function of the fracture aperture. In addition, singularities are present at the fracture front. This usually has a detrimental effect on convergence and stability of coupled models. Coupling between deformations and fluid flow is usually partial (iterative) implicit in hydraulic fracturing simulators. At each time step, several models (geomechanical model, fluid flow model, proppant transport simulator, and thermal solvers) are called sequentially in an iterative loop until convergence is reached. The simulation then advances to the next time step.

3.3

Fracture-matrix fluid exchange (“leakoff”)

We start our journey into the world of coupling in hydraulic fracturing by considering the fluid flow inside a fracture embedded in a porous permeable rock. For a one-dimensional (1D) fracture propagating along the x-axis (PKN or KGD fracture geometry, Fig. 3.3), the flow equation without sources and sinks is given by (Detournay et al., 1990) @q @w 1 1 qL 5 0 @x @t

(3.2)

where t is time; q is the flow rate inside the fracture; w is the fracture width (also sometimes called fracture aperture or fracture opening); qL is the leakoff flow rate through the fracture faces. The leakoff describes how much fluid is lost through the fracture faces into the rock per unit time and per unit length in the direction along the fracture (x-axis). The unit of q is m2/s; the unit of qL is m/s. Both quantities are thereby specified per unit length in the direction normal to page in Fig. 3.3. From Eq. (3.2), qL =2 yields the superficial velocity of the fluid leaking through the fracture face. The factor of 1/2 is due to leakoff occurring through both fracture faces.

Coupling in hydraulic fracturing simulation

51

Figure 3.3 One-dimensional fracture propagating in a permeable porous medium. The direction of fracture propagation is shown with the large gray arrow. The fluid flow inside the fracture and through the fracture faces (leakoff) are shown with small arrows.

For a two-dimensional (2D) fracture located in the xy-plane, the flow equation would read as follows (Abou-Sayed et al., 1984): @qx @qy @w 1 qL 5 0 1 1 @t @x @y

(3.3)

where qx ; qy are the fluid volumes flowing in the x- and y-directions per unit time and per unit length along the y- and x-directions, respectively. The leakoff rate, qL , represents the coupling term between the flow in the fracture and the flow in the adjacent porous media. Eq. (3.3) accounts for the effect leakoff has on the fracture flow. If the porous media flow and fracture flow are two-way coupled, similar leakoff term should enter the porous-media flow equations. In this case, the leakoff flow rate will appear as a boundary condition for the flow and will thus enter the right-hand side in Eq. (3.1b). In hydraulic fracturing models, it is often assumed that qL has the following form (Advani et al., 1990): qL 5


2CL η t2τðxÞ

(3.4)

where CL is the leakoff coefficient; η is the leakoff behavior index; τðxÞ is the arrival time of the fracture tip at location x. Both η and CL are fitting parameters that, in general, can be obtained from filtration experiments for a specific rock. The value of η satisfies η # 1=2 (Wrobel and Mishuris, 2015). If η 5 1=2, the leakoff law is known as Carter’s law: 2CL qL 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 τðxÞ

(3.5)

Carter’s law has been used in hydraulic fracturing models for at least four decades (Clifton and Abou-Sayed, 1979). Leakoff implies that the rock adjacent to the fracture is porous and permeable. It is then intuitively clear that the leakoff rate must depend on the difference between the fluid pressure inside the fracture, P, and the pore pressure inside the rock, Pp .

52

Porous Rock Failure Mechanics

This dependence is ignored in Eqs. (3.4) and (3.5). In order to remedy this shortcoming, Barr (1991) considered the depth of the leakoff zone adjacent to the fracture face, lL . The leakoff flow rate must satisfy Darcy’s law: qL k P 2 Pp 5 μ lL 2

(3.6)

where k is the absolute permeability of the rock; μ is the dynamic viscosity of the leaking fluid. The depth of the leakoff zone is governed by the following equation (Barr, 1991): dlL qL =2 5 ϕ dt

(3.7)

where ϕ is the porosity of the rock. If porosity and permeability do not change over time, Eqs. (3.6) and (3.7) yield (Barr, 1991): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2kϕ P 2 Pp
 qL 5 μ t 2 τðxÞ

(3.8)

Thus, the leakoff coefficient in Eq. (3.5) should depend on the pressure difference between the fracture, and the rock and should be given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi kϕ P 2 Pp CL 5 2μ

(3.9)

The leakoff coefficient in Carter’s law is thus not a material property but depends on the fluid pressures in the fracture and in the porous media around the fracture. The leakoff model given by Eq. (3.4) takes two fitting parameters, CL and η. The leakoff model given Eq. (3.5) takes one fitting parameter, CL . The model given by Eq. (3.8) takes no fitting parameters. The model given by Eq. (3.8) is still a simplification as it neglects the dependence of rock porosity and permeability on time. In reality, as filter cake builds up in the leakoff zone, both porosity and permeability will decrease. Also, using Eq. (3.8) requires that pore pressure values are available. Thus, it can only be applicable if a poromechanical model (for example, Biot’s poroelasticity) is used for the rock surrounding the fracture. Despite its drawbacks, Carter’s law is still widely used today in threedimensional (3D) hydraulic fracturing models, when the fracture is represented as a surface (a collection of surface elements) embedded in a 3D grid of volumetric elements. Likewise, it is used in 2D models when the fracture is represented as a line embedded in a 2D grid. Also, the application of Carter’s law is widespread in 1D models of hydraulic fracturing. In all these models,

Coupling in hydraulic fracturing simulation

53

the dimension of the fracture is effectively lower by one as compared to the dimension of the model. According to Carter’s law, the cumulative leakoff through the fracture faces is given by (Lakhtychkin et al., 2012) QL 5

ðt

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2CL pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds 5 4CL t 2 τðxÞ 1 Sp s 2 τðxÞ τðxÞ

(3.10)

where the integration constant, Sp , is known as spurt loss. Spurt loss is the amount of fluid instantaneously lost through the fracture faces into the formation as soon as the fracture faces are exposed to fluid. Using Eq. (3.10) thus introduces an additional parameter, Sp , into the leakoff model. The value of Sp for a specific combination of fluid and rock can be obtained from laboratory tests. A radically different approach to modeling leakoff is employed when the fracture is represented numerically as a collection of elements of the same spatial dimension as the elements representing the rest of the model. For instance, in a 3D model, the rock can be discretized with tetrahedral or hexahedral finite elements. The fracture can then be represented as a collection of elements (tetrahedra or hexahedra) which meet the tensile or shear failure criteria. These elements can be collectively termed failed elements. In this case, there is no need for an ad-hoc leakoff rule like those of Eqs. (3.4), (3.5), or (3.8) as the flow rate from the fracture into the porous media can be calculated as part of the porous media flow. The permeability and porosity of the failed elements are updated so as to account for the effect of the fracture. This approach has been used for example, in Ji et al. (2009) and Lavrov et al. (2016). A drawback of this approach is the uncertainty about fracture location and orientation within the failed element.

3.4

Coupling fluid and solid

Fluid flow in the fracture is solved to obtain the fluid pressure distribution inside the fracture. This pressure is then used as a boundary condition in the geomechanical solver. The deformations calculated in the geomechanical solver affect the fracture growth and the fracture aperture (opening/closing), and thus affect the fluid flow (Damjanac et al., 2010). This two-way coupling can be illustrated using an example where the fracture is represented using a simple 1D model of a horizontal fracture. Consider coupling from the fracture deformation to the fracture flow. The fluid flow inside the fracture is governed in 1D by Eq. (3.2) which, in discretized form, becomes (Boone and Ingraffea, 1990) ðn11Þ qðn11Þ i11=2 2 qi21=2

Δx

1

wðn11Þ 2 wðnÞ i i 1 qðn11Þ 50 L;i Δt

(3.11)

54

Porous Rock Failure Mechanics

where n is the time step number; i is the discretization cell number along the fracture; Δt is the time step; Δx is the spatial discretization step. The flow rate inside the fracture, for a Newtonian fluid, can be evaluated using the lubrication theory approximation as follows (Brown, 1987; Zimmerman et al., 1991): q52

w3h rP 12μ

(3.12)

where wh is the hydraulic aperture of the fracture. Hydraulic aperture is the aperture of a smooth-walled conduit that would produce the same flow rate under a given pressure gradient as the real rough-walled fracture. For a fracture with perfectly smooth faces, the hydraulic aperture is equal to the aperture of the fracture. For more realistic, rough-walled fractures, the hydraulic aperture is usually smaller (sometimes by a factor of 510) than the mean, “mechanical” aperture of the fracture (the mean, “mechanical” aperture being the average value of the distance between the fracture faces). The difference between the mechanical and the hydraulic apertures is due to the flow tortuosity in roughwalled fractures. The geomechanical solver in a hydraulic fracturing model computes the mechanical aperture. In hydraulic fracturing models, it is often assumed that the hydraulic aperture of the fracture is equal to the mechanical aperture: wh 5 w. [As pointed out by de Pater et al. (1994) and Palmer and Veatch Jr. (1990), the hydraulic aperture being smaller than the mechanical aperture might be amongst the factors responsible for abnormally high fracturing pressures often observed in the field.] Under the assumption of wh 5 w, the flow rates in Eq. (3.11) are given by

qðn11Þ i11=2 5 2

qðn11Þ i21=2 5 2  Factors

 3 wðn11Þ i11=2 Pðn11Þ 2 Pðn11Þ i11

12μ

i

Δx

 3 wðn11Þ i21=2 Pðn11Þ 2 Pðn11Þ i

12μ

Δx

i21

(3.13a)

(3.13b)

   3 3 ðnÞ ðn11Þ ðn11Þ wðn11Þ 2 w and w in Eqs. (3.11), (3.13a) and ; w i i i11=2 i21=2

(3.13b), respectively, effect the coupling from the mechanical deformations to the fracture flow. The coupling in the opposite direction, that is, from the fracture flow to the solid-mechanical model of the porous media, is represented by the leakoff term or by the term LΔt P in Eq. (3.1a) for the pore pressure at the fracture face, which must be equal to the fluid pressure inside the fracture, if filter cake build-up is neglected. In addition, the fluid pressure inside the fracture represents a Neumann boundary condition for the solid-mechanical model of the porous media and thus enters the right-hand side in Eq. (3.1a).

Coupling in hydraulic fracturing simulation

3.5

55

Coupling proppant transport and placement

Proppant transport inside the fracture introduces additional coupling terms into a hydraulic fracturing model. Proppant is transported by the carrying fluid as part of the slurry (proppant particles suspended in the carrying fluid). Thus, the distribution of proppant inside the fracture is affected by the fluid flow. The proppant distribution, in turn, affects the density and viscosity of the slurry. Thereby, the proppant distribution affects the fluid flow inside the fracture. There is thus a two-way coupling between the fluid flow and the proppant transport. The simplest way to introduce proppant transport into a hydraulic fracturing simulator is by using a mixture model. Within this paradigm, the proppant transport reduces to the advection equation for the proppant volume fraction, c. Assuming incompressibility of both the proppant and the carrying fluid, the advection equation is given by [see Adachi et al. (2007) for a detailed discussion of the underlying assumptions] @ðcwÞ 1 r ðcwvp Þ 5 0 @t

(3.14)

where vp is the proppant velocity. The proppant velocity is given by [see for example, Adachi et al. (2007)] vp 5 v 2 ð1 2 cÞvs

(3.15)

where vs is the slip velocity between the carrying fluid and the proppant particles. In the simplest case, the slip velocity is due only to gravity and is thus equal to the hindered settling velocity of proppant particles in the fracture. Several empirical and ad-hoc relations between the magnitude of the settling velocity, vs , and the proppant concentration (and other parameters, such as the fracture width) are available in the literature. For example, the following equation was used in Lakhtychkin et al. (2012):   vs 5 v0 1 2 4:7892c 1 8:8477c2 2 5:918c3

(3.16)

where v0 is the Stokes settling velocity of a single particle. Other examples of vs as a function of c can be found in Gadde et al. (2004) and Clark and Quadir (1981). Eq. (3.14) represents the effect of the carrying fluid on the proppant concentration, that is, coupling from fluid to proppant. Coupling in the opposite direction is represented by the effect proppant has on the fluid flow. In the simplest proppant transport models commonly employed in industrial hydrofrac simulators, this effect is only due to the slurry viscosity being a function of the proppant volume fraction, c. When proppant transport is considered, Eqs. (3.2) and (3.3) should be understood as describing the flow rate of the slurry. The apparent viscosity of the slurry increases with the proppant volume fraction. Moreover, the proppant concentration can affect the type of the slurry rheology: Dilute (c , 0.02)

56

Porous Rock Failure Mechanics

and semi-dilute (0.02 , c , 0.25. . .0.3) suspensions can be approximated as Newtonian fluids (Kuzkin et al., 2013). Concentrated suspensions (c . 0.25. . .0.3) may have a yield stress and thus should be modeled as yield-stress fluids, for example, a Bingham fluid. Even for the least challenging case of dilute and semi-dilute slurries, choosing or constructing the relationship between the slurry viscosity, μ, and the proppant volume fraction, c, is not an easy task (Adachi et al., 2007). Several relationships have been theoretically derived or constructed to fit experimental data for c , 0.25, for instance (Lakhtychkin et al., 2012; Adachi et al., 2007; Shook and Roco, 1991):
 μ 5 μ0 0:9981 1 2:5c 1 10:0c2 1 0:0019 expð20:0cÞ

(3.17a)


 μ 5 μ0 1:0 1 2:5c 1 10:05c2 1 0:00273 expð16:6cÞ

(3.17b)

μ μ 5 0

α c 12 cmax μ μ 5 0

Bcmax c 12 cmax

(3.17c)

(3.17d)

where μ0 is the dynamic viscosity of the carrying fluid (c 5 0); cmax , α, B are fitting parameters that can be determined via laboratory experiments. A critical review and discussion of selected μ versus c relationships can be found in Shook and Roco (1991), Kuzkin et al. (2014), Abedian and Kachanov (2010). One of the problems with using, in particular, Eqs. (3.17c) and (3.17d), is that the slurry viscosity increases rapidly as c ! cmax . As pointed out by Adachi et al. (2007), at such high solids volume fractions, the assumptions underlying the lubrication theory approximation [Eq. (3.12)] are invalidated, and the slurry behaves like a plastic solid material. Another type of coupling involving proppant is from the fracture deformation to the proppant transport. Fracture aperture enters Eq. (3.14) and thus affects the proppant transport. Moreover, if the fracture aperture becomes smaller than a certain threshold, proppant particles will not be able to pass, and proppant transport will stop. This threshold value is related to the proppant particle size. A blocking function based on the proppant particle size was introduced by Dontsov and Peirce (2015) to prevent proppant from entering the fracture tip. Similar approach was taken by Shiozawa and McClure (2016) in their numerical model. The problem is complicated, however, since proppant particles may have a distribution of sizes and may also have irregular shapes. Setting a threshold value for the aperture necessarily implies additional ad-hoc assumptions. In addition to proppant being affected by the fracture deformation (opening/ closing and shear), the proppant itself may affect the fracture deformation once the proppant is deposited in the fracture. In coupled proppant simulations, the normal stress exerted by the proppant on the fracture face is set equal to zero if the proppant concentration is below cmax , and the fracture aperture is above a certain

Coupling in hydraulic fracturing simulation

57

threshold, wth . Concentration above cmax means that a static proppant bed has been formed. Fracture aperture below wth means that proppant particles become immobile (possibly by being embedded into the fracture faces). In both cases, and in their combination, a normal stress is induced on the fracture face. To account for such proppant-induced stresses, an ad-hoc model was constructed by Feng et al. (2016).

3.6

Thermal coupling

Fluids pumped during hydraulic fracturing typically are colder than the formation to be fractured. The fluid’s temperature therefore increases as the fluid travels down the well and into the fracture. This may affect the properties of the fluid. The cold fluid itself would affect the rock by inducing thermal stresses. This makes the in-situ stresses less compressive, which facilitates the fracture opening and reduces the fracturing pressure (Goodarzi et al., 2010). As a result, the aperture of the fracture predicted with a thermo-HM (THM)-coupled model is likely to be larger than that predicted with an HM-coupled model. Given the same injection rate, a THM-coupled model would thus predict a wider and shorter hydraulic fracture (Feng et al., 2016). Shorter fractures and lower fracturing pressures result in smaller total leakoff through the fracture faces in THM-coupled models (Feng et al., 2016). We saw earlier that in an HM-coupled model, fluid pressure in the porous media represents one of the coupling variables, by entering the poroelasticity equations. Fluid pressure in the fracture enters the boundary condition for the fluid flow in the porous media, by governing the fluid flux (leakoff) through the fracture face. Similarly, in a thermo-mechanically coupled model, temperature affects thermal stresses in the porous media and also enters the boundary conditions at the fracture faces. Usually, the thermo-mechanical coupling in porous media is one-way, that is, temperature affects mechanical stresses, but stresses do not affect thermal properties or temperatures (Feng et al., 2016). At the fracture face, thermal coupling between the fracture and the porous media is usually implemented using the convective heat-transfer boundary condition given by (Feng et al., 2016; Holman, 2001)   qn 5 h Tr 2 Tf

(3.18)

where qn is the heat flux through the fracture face; Tr ; Tf are temperatures of the rock and of the fluid inside the fracture, respectively; h is the convective heattransfer coefficient. In addition to the thermo-mechanical coupling in the porous media (the temperature affecting the stresses), there is a thermo-hydraulic coupling (the temperature affecting the pore pressure). Both thermo-mechanical and thermo-hydraulic couplings can be taken into account in thermo-HM-coupled poroelasticity, see for example, Wang and Papamichos (1994) for details.

58

3.7

Porous Rock Failure Mechanics

Coupling in acid fracturing

Some additional types of coupling need to be added in numerical models of acid fracturing. First, acid solution is transported in the fracture by the fluid flow. Thus, there is coupling from the flow in the fracture to the acid transport. The fluid velocity enters the mass balance equation for the acid solution and thus provides the coupling. For instance, in a planar 2D fracture [cf. Eq. (3.3)], the mass balance equation for the acid transport in the plane of the fracture would be given by (Dong et al., 2002)       @ Cqx @ Cqy @ Cw 1 CqL 1 2Ckg 5 0 1 1 @t @x @y

(3.19)

where C is the acid concentration, C, averaged along the fracture width at a given location in the fracture plane (x, y); kg is the apparent mass-transfer coefficient defined as follows (Dong et al., 2002):     @C  5 kg C 2 Cz5w=2 2D (3.20) @z z5w=2 where D is the acid diffusion coefficient; the z-axis points normal to the fracture face, and z 5 w=2 is located at the face. Second, another coupling exists, being due to the etching effect that alters the aperture of the fracture. At a given location (x, y) in the fracture plane, acid is transported normally to the fracture face (in the z-direction) by two mechanisms: leakoff and diffusion (Settari, 1993). While acid diffusion may take place over the entire fracture face, acid leakoff occurs in form of wormholing. As a result, only a fraction, η, of the leaked-off acid contributes to the fracture-aperture enhancement. Hence, the effect of acid transport on the fracture geometry can be described as follows (Oeth et al., 2013; Dong et al., 2002):   @w β 5 2kg C 1 ηCqL @t ρð1 2 ϕÞ

(3.21)

where β represents the dissolving power of the acid; ρ and ϕ are the density and porosity of the rock, respectively. Acid fracturing thus involves a two-way coupling: acid etches the fracture faces and, as a result, the fracture aperture is affected according to Eq. (3.21). The fracture aperture, in turn, affects the fluid flow in the fracture, which affects the acid transport, and so forth.

3.8

Conclusion

An overview of the most important types of coupling in hydraulic fracturing simulations has been presented in this chapter. These are summarized in Fig. 3.4. The entanglement of different numerical models, one affecting another, evident

Figure 3.4 Coupling of thermal, hydraulic, and mechanical processes in a hydraulic fracturing simulator. Boxes represent models. Italic labels indicate variables passed between the coupled models. Arrows indicate the directions of data transfer between the models.

60

Porous Rock Failure Mechanics

from Fig. 3.4, introduces extra complexity in hydraulic fracturing simulations. This complexity comes in addition to each model already being quite complicated and numerically challenging, both in terms of computational speed and computational stability. The necessity to run a coupled model until convergence is reached increases the computational time and may also introduce extra instabilities solely due to coupling. Continuous progress in numerical methods, solution techniques, and computer hardware enable increasingly more advanced and accurate simulations of hydraulic fracturing, with 3D models gradually paving their way in the industry. This trend is expected to continue in the future, resulting in better design of stimulation jobs and overall more efficient and cost-effective treatments.

References Abedian, B., Kachanov, M., 2010. On the effective viscosity of suspensions. Int. J. Eng. Sci. 48, 962965. Abou-Sayed, A.S., Sinha, K.P., Clifton, R.J. 1984. Evaluation of the Influence of In-Situ Reservoir Conditions on the Geometry of Hydraulic Fractures Using a 3-D Simulatort: Part 1—Technical Approach. SPE/DOE/GRI Paper 12877 Presented at the 1984 SPE/DOE/GRI Unconventional Gas Recovery Symposium held in Pittsburgh, PA, May 1315, 1984. Adachi, J., Siebrits, E., Peirce, A., Desroches, J., 2007. Computer simulation of hydraulic fractures. Int. J. Rock Mech. Mining Sci. 44, 739757. Advani, S.H., Lee, T.S., Lee, J.K., 1990. Three-dimensional modeling of hydraulic fractures in layered media: Part I—Finite element formulations. Trans. ASME J. Energy Resour. Technol. 112, 19. Barr, D.T., 1991. Leading-Edge Analysis of Correct Simulation of Interface Separation and Hydraulic Fracturing. PhD Dissertation., MIT. Boone, T.J., Ingraffea, A.R., 1990. A numerical procedure for simulation of hydraulicallydriven fracture propagation in poroelastic media. Int. J. Numerical Analyt. Methods Geomech. 14, 2747. Brown, S.R., 1987. Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. B. 92, 13371347. Clark, P.E., Quadir, J.A., 1981. Prop Transport in Hydraulic Fractures: A Critical Review of Particle Settling Velocity Equations. SPE/DOE Paper 9866 Presented at the 1981 SPE/DOE Low Permeability Symposium held in Denver, Colorado, May 2729, 1981. Clifton, R.J., Abou-Sayed, A.S., 1979. On the computation of the Three-Dimensional Geometry of Hydraulic Fractures. SPE Paper 7943 Presented at the Symposium on Low Permeability Gas Reservoirs, Denver, Colorado, 2022 May, 1979. Damjanac, B., Gil, I., Pierce, M., Sanchez, M., Van As, A., Mclennan, J., 2010. A new approach to hydraulic fracturing modeling in naturally fractured reservoirs. ARMA Paper 10-400. The 44th US Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, Held in Salt Lake City, UT, June 2730, 2010. de Pater, C.J., Cleary, M.P., Quinn, T.S., Barr, D.T., Johnson, D.E., Weijers, L., 1994. Experimental verification of dimensional analysis for hydraulic fracturing. SPE Product. Facil. 9, 230238.

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Detournay, E., Cheng, A.H.-D., Mclennan, J.D., 1990. A poroelastic PKN hydraulic fracture model based on an explicit moving mesh algorithm. Trans. ASME J. Energy Resour. Technol. 112, 224230. Dong, C., Zhu, D., Hill, A.D., 2002. Modeling of the acidizing process in naturally fractured carbonates. SPE J. 7, 400408. Dontsov, E.V., Peirce, A.P., 2015. Proppant transport in hydraulic fracturing: crack tip screen-out in KGD and P3D models. Int. J. Solids Struct. 63, 206218. Doster, F., Nordbotten, J.M., 2015. Full pressure coupling for geo-mechanical multi-phase multi-component flow simulations. SPE Paper 173232 Presented at the SPE Reservoir Simulation Symposium Held in Houston, Texas, USA, 2325 February 2015. Feng, W., Were, P., Li, M., Hou, Z., Zhou, L., 2016. Numerical study on hydraulic fracturing in tight gas formation in consideration of thermal effects and THM coupled processes. J. Petroleum Sci. Eng. 146, 241254. Gadde, P.B., Liu, Y., Norman, J., Bonnecaze, R., Sharma, M.M., 2004. Modeling proppant settling in water-fracs. SPE Paper 89875 Presented at the SPE Annual Technical Conference and Exhibition Held in Houston, Texas, USA, 2629 September 2004. Goodarzi, S., Settari, A., Zoback, M., Keith, D.W., 2010. Thermal aspects of geomechanics and induced fracturing in CO2 injection with application to CO2 sequestration in Ohio River Valley. SPE Paper 139706 Presented at the SPE International Conference on CO2 Capture, Storage, and Utilization Held in New Orleans, Louisian, USA, 1012 November 2010. Holman, J.P., 2001. Heat Transfer. McGraw-Hill, Singapore. Ji, L., Settari, A., Sullivan, R.B., 2009. A novel hydraulic fracturing model fully coupled with geomechanics and reservoir simulation. SPE J.423430. Kuzkin, V.A., Krivtsov, A.M. & Linkov, A.M. 2013. Proppant transport in hydraulic fractures: computer simulation of effective properties and movement of the suspension. Proceedings of XLI International Summer School—Conference APM 2013, pp. 322337. Kuzkin, V.A., Krivtsov, A.M., Linkov, A.M., 2014. Computer simulation of effective viscosity of fluid-proppant mixture used in hydraulic fracturing. J. Mining Sci. 50, 19. Lakhtychkin, A., Eskin, D., Vinogradov, O., 2012. Modelling of transport of two proppantladen immiscible power-law fluids through an expanding fracture. Can. J. Chem. Eng. 90, 528543. Lavrov, A., Larsen, A., Bauer, A., 2016. Coupling a fracturing code to a transient reservoir simulator: a hands-on approach. ARMA Paper 16-560 Presented at the 50th US Rock Mechanics/Geomechanics Symposium Held in Houston, Texas, USA, 2629 June 2016. Longuemare, P., Mainguy, M., Lemonnier, P., Onaisi, A., Ge´rard, C., Koutsabeloulis, N., 2002. Geomechanics in reservoir simulation: overview of coupling methods and field case study. Oil Gas Sci. Technol.—Rev. IFP. 57, 471483. Oeth, C.V., Hill, A.D., Zhu, D., 2013. Acid fracturing: fully 3D simulation and performance prediction. SPE Paper 163840 Presented at the SPE Hydraulic Fracturing Technology Conference Held in The Woodlands, Texas, USA, 46 February 2013. Palmer, J.D., Veatch Jr, R.W., 1990. Abnormally high fracturing pressures in step-rate tests. SPE Prod. Eng. 5, 315323. Schrefler, B.A., Scotta, R., 2001. A fully coupled dynamic model for two-phase fluid flow in deformable porous media. Comput. Methods Appl. Mech. Eng. 190, 32233246. Settari, A., 1993. Modeling of acid-fracturing treatments. SPE Prod. Facilities. 8, 3038. Settari, A., Walters, D.A., 2001. Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. SPE J. 6, 334342.

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Shiozawa, S., McClure, M., 2016. Simulation of proppant transport with gravitational settling and fracture closure in a three-dimensional hydraulic fracturing simulator. J. Petroleum Sci. Eng. 138, 298314. Shook, C.A., Roco, M.C., 1991. Slurry Flow: Principles and Practice. ButterworthHeinemann, Stoneham. Wang, Y., Papamichos, E., 1994. Conductive heat flow and thermally induced fluid flow around a well bore in a poroelastic medium. Water. Resour. Res. 30, 33753384. Wrobel, M., Mishuris, G., 2015. Hydraulic fracture revisited: particle velocity based simulation. Int. J. Eng. Sci. 94, 2358. Zimmerman, R.W., Kumar, S., Bodvarsson, G.S., 1991. Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 28, 325331.

Stress-induced permeability evolutions and erosion damage of porous rocks

4

Dawei Hu1, Hui Zhou1, Jianfu Shao2 and Fan Zhang3 1 Institution of Rock and Soil Mechanics, Chinese Academy of Sciences, Beijing, China, 2 University of Lille 1, Villeneuve d’Ascq, France, 3Hubei University of Technologies, Wuhan, China

4.1

Introduction

Geo-engineering applications, such as oil and gas production, geological storage of nuclear wastes, sequestration of carbon and residual gas, usually encounter underground water during their construction and operation period. Rock materials (sandstone, limestone, granite, etc.) exhibit stress-induced damage due to nucleation and propagation of microcracks (Tapponier and Brace, 1976; Wong, 1982; Moore and Lockner, 1995; Wong et al., 1997; Baud et al., 1999; Xu et al., 2015). The main consequences of the induced damage including nonlinear stress
strain relations, degradation of elastic properties and induced anisotropy, volumetric dilatation, material softening, and permeability variations (Ismail and Murrell, 1976; Steif, 1984; Horii and Nemat-Nasser, 1985; Fredrich et al., 1989; Zhu and Shao, 2015). A number of laboratory investigations have contributed to the evaluation of permeability during rock damage and cracking, for instance (Zhu and Wong, 1997; Suzuki et al., 1998; Schulze et al., 2001; Souley et al., 2001; Bossart et al., 2002; Wang and Park, 2002; Oda et al., 2002; Shao et al., 2005; Zhu et al., 2016). These works have clearly shown that the rock permeability evolution is directly related to the distribution, opening, and coalescence of induced microcracks. However, few investigations (Hu et al., 2010a; Jiang et al., 2010) indicated that pore compaction of porous rocks under applied stress could result in permeability reduction, and generated competition with permeability increase induced by microcrack propagation. Moreover, the structure of geo-engineering applications is subjected not only to mechanical loading but also to chemical degradation when they contact with aggressive fluid flow, for example, acid rain from atmosphere, carbonic acid in karst region, CO2 injection in CO2 geological reservoir, and so on. The fluid
rock interactions may cause a chemical and flow transport regime disturbance, which will result in the variation of the porosity, permeability, and mechanical stability of rock material in the long term (le Guen et al., 2007; Orlic and Wassing, 2013; Zhou et al., 2016). Previous investigations (Spiers and Schutjens, 1990; Lehner, 1990; Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00004-X © 2017 Elsevier Ltd. All rights reserved.

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Seto et al., 1997; Homand and Shao, 2000; Li et al., 2003; Wang and Wang, 2004; Rimmele´ et al., 2010; Fernandez-Merodo et al., 2007; Xie et al., 2011, Jiang et al., 2012; Bond et al., 2013) have shown that this phenomenon is a fully hydromechanical
chemical (HMC) coupling problem. In one hand, the chemical degradation induces an increase of material porosity, which will cause the deterioration of mechanical properties (i.e., diminution of elastic modulus and strength) and accelerate the transport phenomena (i.e., increase of permeability and diffusion). On the other hand, mechanical loading generates microcracks or pore compactions in geomaterials, which will change the path of mass transfer and thereby change the rate of chemical degradation. A correct understanding and prediction of these mechanisms, therefore, is necessary for the performance of fully coupled HMC numerical analyses. Different constitutive models (Wang and Wang, 2004; Xie and Shao, 2006; Lydzba et al., 2007; Pietruszczak et al., 2006; Carde et al., 1997; Taron et al., 2009; Zhao et al., 2011; Choi et al., 2004) have been proposed to describe this HMC coupling behavior of geomaterial. These models deal with the HMC coupling problem by coupling chemistry, transport, and mechanics with the help of both phenomenological approach on the macroscale and micromechanical approach on the multiscale. In these models, the main work is often focused on mechanical modeling, and the chemical reactions are empirically formulated to investigate their influence on mechanical properties. Therefore, these models cannot take into account the kinetics of chemical reactions of geomaterial. In this chapter, the competition between pore compaction-induced permeability decrease and crack-induced permeability increase of sandstone is investigated during triaxial compression by both steady and transient methods, and HMC coupling tests of sandstone are performed to investigate the evolution of the creep strain, permeability, and elastic modulus with CO2
brine
rock interaction. A general framework of HMC coupling model is proposed for geomaterial taking into account the coupling effect of stress-induced damage, chemical damage, and permeability evolution. The numerical algorithm and identification of model parameters as well as the numerical results are given in the last section.

4.2

Laboratory tests

4.2.1 Steady and transient permeability tests of sandstone under triaxial compression Two different experimental techniques are usually used for the measurement of rock permeability: steady-state flow (or permanent regime) method, and pulse test (or transient regime) method. The choice of the one or the other method mainly depends on the range of permeability to be determined. Generally, for materials with relatively higher permeability (e.g., .10216 m2), it is easy to reach the permanent flow regime in the sample and the steady-state flow is preferred. However, for rocks with low and very low permeability, the set-up of the steady-state flow needs a very long time period and it is then technically impossible to directly estimate the

Stress-induced permeability evolutions and erosion damage of porous rocks

65

permeability. An indirect method, the most largely used one is called the pulse test method (Brace et al., 1968; Bear, 1972), is needed. The permeability is estimated from pressure evolution with time, using an inverse numerical algorithm. In this section, steady and transient methods were respectively performed to measure the permeability of two different sandstone, for example, moderate porosity sandstone (porosity of 21%) and low porosity sandstone (porosity of 5%), the permeability evolutions were related to competition effect of pore compaction and microcrack propagation caused by triaxial stress applied.

4.2.1.1 Steady permeability tests The rock studied for steady permeability tests is red sandstone with an average porosity of 21%. The petrophysical analysis with the X-ray diffraction technique shows that the main mineral compositions of the sandstone are quartz, feldspar, mica, and calcite. The microscopic analysis by SEM technique shows that quartz and feldspar grains are ovoid and surrounded by mica and calcite; this results in a continuous pore network that constitutes the connected porosity for interstitial fluid flow. In the case of the sandstone studied here, its initial permeability is relatively high and estimated to be about 10216 m2 . In this regard, the steady-state flow method was then applied. A schematic illustration of the testing procedure is shown in Fig. 4.1. It consists in the injection of water from the bottom surface of the sample in order to obtain an incremental variation of the interstitial pressure ðΔpÞ on this surface, while keeping the pressure at the upper surface constant. When the steady-state flow is established, the pressure variation ðΔpÞ and the injection flow rate, noted as

Outlet σ1

σ2 = σ3 = Pc L

σ1 Δp Injection

Figure 4.1 Schematic illustration of steady permeability test.

66

Porous Rock Failure Mechanics

Qðm3 =sÞ, become constant in elapsed time. Applying the classic Darcy’s law, the intrinsic permeability (noted as k) can be easily deduced by as follows:
 QμL k m2 5 ΔpA

(4.1)

The coefficient μ denotes the dynamic fluid viscosity coefficient and equals μ 5 1:005 3 1023 Pa s under the room temperature; L and A are the length and cross-section area of the sample, respectively. The rock mechanics sign convention is used throughout this chapter. A positive sign is used for compressive stresses and strains. Furthermore, a fixed coordinate frame is used for the cylinder sample, and the cylinder axis was parallel to the x1 axis. σi and εi (i 5 1; 2; 3) denote the three principal stresses and strains in this frame. The typical stress
strain curves obtained in these triaxial tests for permeability measurement are plotted in Fig. 4.2. We observe the induced degradation of elastic moduli, transition from volumetric compressibility to dilatation, and the transition from brittle to ductile behavior with the confining pressure. Fig. 4.2 shows the variations of the intrinsic permeability with the relative axial strain (ε1 =εpeak 1 ) during the triaxial compression tests with the four confining pressures of 5, 10, 20, and 30 MPa. Note that only the permeability in the axial direction is measured in the present work due to technical limitation of the device. However, as the induced microcracks are mainly oriented in the axial direction, the permeability in this direction should be more significantly affected than that in the radial direction. From these curves under the different confining pressures, the most important feature of the axial permeability variation seems to be that the permeability decreases rather quickly during the two first stages of rock deformation, say the closure phase of pore compaction and the linear elastic deformation phase. After that, with the onset of propagation of induced microcracks, the diminution of permeability is attenuated and an increase of permeability is even observed for low confining pressures. Such results seem to indicate that due to the relatively high value of the initial permeability of the sandstone, the variation of the permeability is more sensitive to the closure of pore compaction than to the growth of induced damage. The effect of induced damage on the sandstone permeability becomes dominate only at the late stage of induced damage approaching to the coalescence of microcracks. At the diffuse regime of damage, the sandstone permeability is very moderately affected. As for the permeability variation during the unloading
reloading cycle, the permeability increases slightly during the unloading stage and then decreases significantly during the reloading stage (Fig. 4.3). The permeability at the end of unloading does not recover its initial value before loading. All these phenomena indicate that the diminution of permeability is related to some irreversible deformation process.

4.2.1.2 Transient pulse tests The pulse test method was developed by Brace et al. (1968) for low permeability rocks. A schematic illustration of the testing procedure is shown in Fig. 4.4.

Stress-induced permeability evolutions and erosion damage of porous rocks

30 20 σ2 = σ3 = 5MPa 0.4

0.8

0

1.2

0.2

0.4

σ1-σ3 (MPa)

2.7

40 30 20 10

σ2 = σ3 = 10MPa

0 –1.6 –1.2 –0.8 –0.4

0.4

0.8

1.2

0 2.3 –0.5 1.9 –1

σ2 = σ3 = 10MPa 0

1.6

0.2 0.4 0.6 0.8

Permeability (10–16m2)

1.8

0.4

1.4

0.2

1.2

0

1 0.8

0.8

1.2

0

1.6

0.2 0.4 0.6 0.8

–0.4 1.2 1.4

40 σ2 = σ3 = 30MPa

0

Permeability (10–16m2)

1.8

60

1

ε1/ε1peak

σ1-σ3 (MPa)

80

–1.2 –0.9 –0.6 –0.3

–0.2

σ2 = σ3 = 20MPa

0.6 0.4

Permeability Volume strain

1.6

0.3 0.6 0.9 1.2 1.5 1.8 ε1 (%)

0.8 0.6

1.4

0.4

1.2 0.2

1 0.8

0

σ2 = σ3 = 30MPa

–0.2

0.6 0

0.6

1.6

ε1 (%)

20

–1.5 1.2 1.4 1.6

Permeablility Volume strain

2

σ2 = σ3 = 20MPa

ε3 (%)

100

1

ε1/ε1peak

σ1-σ3 (MPa)

0

–1.2 1.2

1

Permeability 0.5 Volume strain

ε1 (%)

–0.4

0.8

1.5

0

ε3 (%) 90 80 70 60 50 40 30 20 10 0

0.6

ε1/ε1peak

50

ε3 (%)

σ2 = σ3 = 5MPa

ε1 (%)

60

–0.8

–0.8

2.4

Volume strain (%)

0

ε3 (%)

–1.2

–0.4 2.6

Volume strain (%)

–0.4

Permeability (10 –16m2)

–0.8

0

2.2

0 –1.2

3 2.8

volume strain (%)

10

Permeability 0.4 Volume strain volume strain (%)

40

Permeability (10–16m2)

3.2

σ1-σ3 (MPa)

50

67

0

0.2

0.4

0.6

0.8

1

1.2

ε1/ε1peak

Figure 4.2 Stress
strain curves, variations of permeability, and volumetric deformation during triaxial tests for permeability measurement.

The studied rock for the transient pulse tests is fresh sandstone without weathering, and has a tight pore network with low porosity of about 5%. Two reservoirs were connected to the both ends of the sample, and the volumes of the upstream and downstream reservoirs are, respectively, 100 mL and 50 mL.

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Permeability (10–16m2)

3.5 5MPa 10MPa 20MPa 30MPa

3 2.5 2 1.5 1 0.5

0

20

40 60 Mean stress (MPa)

80

Figure 4.3 Relationship between intrinsic permeability and effective mean stress during triaxial compression tests with different confining pressure.

p2, V2 Downstream

σ1

σ3

L

Sample q σ1 Upstream p1, V1

Figure 4.4 Schematic illustration of transient pulse test.

After the loading of confining pressure on the samples, deviatoric stress was applied with several repeated loading
unloading cycles. During one loading cycle, the pulse test was performed after the sample deformation reaches stable. A given pore pressure was consequently applied on both ends of the sample. When pressure equilibrium was achieved in the reservoir
sample
reservoir system, a step increase in pressure was applied to the upstream reservoir. An exponential decay of pressure difference across the sample was monitored as a pressure pulse diffused through the sample; the pressure
time histories of both reservoirs were then monitored until

Stress-induced permeability evolutions and erosion damage of porous rocks

69

pressure equilibrium in the reservoir
sample
reservoir system was re-achieved. The permeability of the samples was calculated from the slope of the decay curve based on the following equation (Brace et al., 1968): pi 2 pf 5 Δp

   V2 1 V2 e2αt V1

(4.2)

with 

kA α5 μβL



1 1 1 V1 V2

 (4.3)

A and L are respectively the cross-sectional area and length of the sample, V1 and V2 are respectively the volumes of the upstream and downstream reservoirs, pi and pf are respectively the initial and final equilibrium pressure, and Δp is the step increase of pressure in upstream reservoir at time 5 0. β is water compressibility and is about 4.5 3 1024 MPa21. The typical stress
strain curves of triaxial compression tests with confining pressure of 5 MPa and 15 MPa are presented in Fig. 4.5. The similar mechanical responses, for example, nonlinear stress
strain relation and the induced degradation of elastic moduli, are observed as that in Fig. 4.2. The permeability evolutions of triaxial compression test with confining pressure of 5 MPa and 15 MPa are illustrated in Fig. 4.6. This low porosity sandstone has a relatively low permeability, that is, 10217
10218 m2, which is 1
2 orders of magnitude smaller than the one of moderate porosity sandstone in Section 2.1.1. However, the permeability evolution shows the similar trends, for example, permeability reduction due to pore compaction, permeability increase due to microcrack propagation, as shown in Fig. 4.2.

4.2.2 Hydro-mechanical
chemical coupling behavior of sandstone Two kinds of laboratory test, for example, creep tests with injection of CO2 alone and CO2
brine solution, indentation test on samples after CO2
brine
rock reaction, were performed on the moderate porosity sandstone, the evolutions of elastic modulus, creep strain, and permeability were analyzed and related to the hydrostress
chemical coupling effects.

4.2.2.1 Creep tests with injection of CO2 alone and CO2
brine solution The creep tests were conducted with a thermal
hydro-mechanical and chemical (THMC) coupling testing system under conventional triaxial conditions. A schematic illustration of the testing cell is shown in Fig. 4.7. Moreover, an autoclave

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Porous Rock Failure Mechanics

80

σ1– σ3 (MPa)

70 60 50 40 30

Pc = 5MPa

20 10

ε1 (%)

0 0

0.2

0.4

0.6

0.8

1

σ1– σ3 (MPa)

120 100 80 60 40

Pc = 15MPa

20

ε1 (%)

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 4.5 Typical stress
strain curves of triaxial compression tests with different confining pressure.

was used to inject CO2 alone or CO2
brine solution into the samples. The autoclave has one upper outlet and one lower outlet for the injection of CO2 alone or CO2
brine solution, respectively. The whole testing system is then placed into an oven to perform the tests at a given temperature. The testing room is also equipped with an air conditioner to maintain a constant temperature of 20 6 2 C. Consequently, the temperature condition of the tests can be maintained at a precision of 6 0.2 C. The sample was first saturated with NaCl solution (0.01 mol/L) under vacuum conditions before each test. The moderate saline solutions were similar to natural aquifers (le Guen et al., 2007). The sample was inserted inside a rubber jacket and was thus isolated from the confining pressure fluid. The sample was then placed between two porous steel pads to ensure a uniform distribution of fluid pressure at the inlet and outlet faces of the sample. Limited data are available with respect to the in situ stress and temperature of the eastern basin. For simplicity, we used the linear approximations of the stress gradient and temperature gradient reported by Qiao et al. (2012). The values that

Stress-induced permeability evolutions and erosion damage of porous rocks

71

K (m2)

4.4E-17

Pc = 5MPa

4.2E-17 4E-17 3.8E-17 3.6E-17 3.4E-17 3.2E-17

σ1–σ3(MPa)

3E-17 0

10

20

30

40

50

60

70

80

(A) Confining pressure of 5 MPa K (m2)

9E-18 8.5E-18

Pc = 15MPa

8E-18 7.5E-18 7E-18 6.5E-18 6E-18 σ1–σ3(MPa)

5.5E-18 5E-18 0

20

40

60

80

100

120

(B) Confining pressure of 15 MPa

Figure 4.6 Permeability evolutions of triaxial compression test with confining pressure of 5 and 15 MPa. (A) Confining pressure of 5 MPa, (B) Confining pressure of 15 MPa.

Oven

Deviatoric stress Injection of NaCl solution Confining stress

CO2 LVDT Ring Sample NaCl solution

Figure 4.7 THMC coupling testing system.

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we used for the temperature gradient and vertical stress gradient are 30.8 C/km and 20.5 MPa/km, respectively. Assuming a temperature and stress at zero depth of 15 C and 0 MPa, respectively, the temperature and vertical stress at an aquifer depth of 800 m are approximately 40 C and 16.4 MPa, respectively. Therefore, the present experiments were conducted at a temperature of 40 C, and the confining stress and axial stress were loaded to 12 MPa and 17 MPa, respectively. These temperature and stress conditions are similar to those of natural aquifers. The confining stress (σ1 5 σ2 5 σ3 ) was first loaded to 12 MPa at a rate of 0.1 MPa/s, which was sufficiently low for the drained condition. After the application of the confining pressure, the saturation of the sample was verified by the injection of NaCl solution (0.01 mol/L) from one side face until the uniform pressure on the other side face reached equilibrium. The deviatoric stress (σ1 2 σ3 ) was applied and held at 5 MPa to conduct the creep test.

Creep tests with injection of CO2 alone When the creep strain under a confining stress of 12 MPa and an axial stress of 17 MPa reached stability, pure CO2 was injected from the upper outlet of the autoclave into the sample in two cases. In the first case, the pressures at the inlet and outlet were equal to 3 MPa and 2.5 MPa, respectively. A low-pressure drop of 0.5 MPa was created to generate a low CO2 flow through the sample. After the stabilization of the axial and lateral deformation induced by the injection of lowpressure CO2, the pressures at the inlet and outlet were increased to 8 MPa and 7.5 MPa, respectively. The triple and critical points of pure CO2 are 0.518 MPa at 256.6 C and 7.35 MPa at 31 C; thus, the CO2 injected at 3 and 8 MPa was in the gas phase and supercritical phase, respectively. The experiments started with samples in a saturated state (no CO2), and the creep strain stabilized after 48 h. No sufficient steady-state axial and lateral strain was observed in the absence of CO2. Immediately after the first injection of CO2 in the gas phase, both the axial and lateral strains exhibited an instantaneous strain response, followed by strain rates that significantly decreased with time and stabilized after 72 h. The instantaneous strain response had a volumetric dilation and was attributed to the loading of the pore pressure. According to the effective stress concept, an increase in the pore pressure can cause a decrease in the effective confining stress and result in volumetric dilation. A similar strain response was observed with the injection of CO2 in the supercritical phase. Examining the creep strain evolution in detail (Fig. 4.8), the instantaneous volumetric dilation and the creep strain, which were caused by CO2 in the gas and supercritical phases, had the same magnitude, and the ratio of the instantaneous strain variation between the gas and supercritical phases was approximately 1 vs 3, which nearly coincided with the ratio of the pore pressure (3 vs 8 MPa). Therefore, the variation of the effective confining stress caused by the injection of CO2 alone may be assumed to play a dominant factor in the strain response, and the CO2 injected in both the gas and supercritical phases is chemically inert to rock in the present work.

Stress-induced permeability evolutions and erosion damage of porous rocks

73

Creep tests with injection of CO2
brine

Creep tests with the injection of CO2
brine were also conducted by the THMC coupling testing system, and the stress state and temperature were the same as in the creep tests with the injection of CO2 alone. However, CO2
brine with a constant pressure of 8 MPa was injected into the sample from the lower outlet of the autoclave. The outlet of the sample was held at 7.5 MPa. Fig. 4.9 presents the evolution of the creep strain during the creep test with the injection of CO2
brine. After the injection of CO2
brine, an instantaneous strain response was observed due to the change in the effective confining stress. Consequently, both the axial strain and lateral strain underwent clear primary and secondary creep stages. The steady-state creep strain rates in the axial and lateral 3MPa CO2 gas

0.8 0.6

8MPa CO2 supercritical

Strain (10–3)

0.4 Axial Lateral

0.2 0 0

40

80

–0.2

120 160 200 240 280 320 360 400 Time (hour)

–0.4 –0.6

Figure 4.8 Evolution of strain during the creep tests with injection of CO2 alone.

Strain (10–3)

1.2 0.8 0.4 0 –0.4

0

100

200

300

400 500 Time (hour)

–0.8 –1.2

Axial

–1.6

Lateral

–2 –2.4

Figure 4.9 Evolution of strain in the creep tests with injection of CO2
brine.

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Porous Rock Failure Mechanics

1E-15

0

100

200

300

400

500

Time (hour)

1E-16

Permeability (m2) 1E-17

Figure 4.10 Evolution of permeability after the injection of CO2
brine.

directions were ε_ 1 5 2:7 3 10210 s21 and ε_ 3 5 2 5:4 3 10210 s21 . Similar results have been reported by le Guen et al. (2007) during flow-through experiments on limestone and sandstone, as well as by other researchers with aggregates of different mineralogies (Zhang and Spiers, 2005). The evolution of the permeability was also measured during the injection of CO2
brine. Due to the difficulty of measuring the weight, a flow ratio meter with a precision of 0.001 mL/min was used to measure the flow ratio. When the steadystate flow was established, the injection flow rate can be recorded and is assumed to be constant over a short time period. Applying the classic Darcy’s law, the intrinsic permeability can be calculated. According to the viscosity models of H2O 1 NaCl 1 CO2 proposed by Fleury and Deschamps (2008), the viscosity coefficient of the CO2
brine used in this study was calculated to be approximately 0.82 3 1024 Pa s. Fig. 4.10 shows the evolution of the permeability after the injection of CO2
brine. The permeability immediately decreased from its initial value of 6.6 3 10216
1.1 3 10216 m2 during the primary creep stage. The permeability continued to decrease at a constant rate of approximately 29.6 3 10220 m2/h. According to the previous geochemical experiments (Rosenbauer et al., 2005), the CO2
brine
rock reaction causes an increase in the porosity, which may enlarge the transport channels in the rock matrix and lead to increased permeability. However, the evolution of the permeability in the present study showed the opposite tendency and may be attributed to the compaction of the pore space under stress effects and the generation of preferential channels (not connected to the two end faces of sample) induced by the CO2
brine
rock reaction (Izgec et al., 2008).

4.2.2.2 Indentation tests on samples after CO2
brine
rock reaction Indentation tests were performed to study the evolution of the elastic modulus of the samples after the CO2
brine
rock reaction. The samples (cylindrical specimens 37 mm in diameter and 5 mm in height) were positioned at the base of the autoclave and were covered with 0.01 mol/L NaCl solution. CO2 was then injected into the autoclave to reach a pressure of 8 MPa. The autoclave was placed into an

Stress-induced permeability evolutions and erosion damage of porous rocks

75

incubator, and the temperature was set to 40 C. After different reaction times, one reacted sample was removed from the autoclave to perform the indentation tests. A specific testing device (named MICROPE) was used to perform the indentation tests (Zhang et al., 2012). During the test, a cylindrical indenter under the applied load with a constant displacement rate (0.073 mm/min) penetrated into the rock surface, forming a crater, and the penetration displacement vs applied force curves were recorded. The typical result of an indentation test was represented by the penetration displacement (e) curve vs the applied force (P). Assuming that the material under the indenter remains in the elastic domain and based on the classical solution for the penetration problem in an infinite isotropic elastic body, the slope of the penetration curve can be related to the elastic properties of the material (Boussinesq, 1885): dP ED 5 de 1 2 v2

(4.4)

where E denotes the elastic modulus, v is Poisson’s ratio, and D is the diameter of the indenter. According to Eq. (4.4), two unknown elastic parameters must be determined from one experimental curve. An additional assumption is then needed. Poisson’s ratio appears in a square form in Eq. (4.4), and a simple error analysis shows that its influence is generally smaller than that of elastic modulus. Therefore, the value of Poisson’s ratio is fixed to a constant value a priori in view of the determination of Young’s modulus. According to the previous triaxial compression tests (Hu et al., 2010a), the value of Poisson’s ratio was chosen as 0.2 in this study. Fig. 4.11 demonstrates the typical force
penetration curves of the indentation tests on samples after different reaction times, including 0, 3, 15, 30, 60, and 240 days. The force
penetration curves were significantly influenced by the fluid
rock reaction. Both the elastic modulus and force corresponding to the plastic yield 250

Force (N)

200 0 day

150

3 days 15 days

100

30 days 60 days

50

240 days

0 0

0.05

0.1

0.15

0.2

0.25

Penetration (mm)

Figure 4.11 Typical force
penetration curves of indentation tests on samples after different periods of reaction time.

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Porous Rock Failure Mechanics

8500

E (MPa)

8000 7500 7000 6500 6000 5500 5000 0

50

100 150 Time (day)

200

250

Figure 4.12 Evolution of elastic modulus of modulus after different periods of reaction time.

threshold decreased with an increasing reaction time. This finding corresponds with the previous observations of the variation of the tensile strength of Bentheimer sandstone aged in CO2-saturated salt water (Ojala, 2011). For clarity, we report the evolution of the average values of the elastic modulus as a function of the reaction time in Fig. 4.12. The elastic modulus immediately decreased in the initial periods, for example, 3 days and 15 days. After the initial periods, the rate of decrease was attenuated and approached a constant value of approximately 21.35 MPa/day. The evolution of the elastic modulus had the same trend as the evolution of the creep deformation, as shown in Fig. 4.9. The degradation of the elastic modulus induced by the chemical reaction confirmed with the previous experimental results on the studied sandstone (Cui et al., 2008; Feng et al., 2004, 2009) and was similar to the leaching phenomenon in cement-based material (Heukamp et al., 2001).

4.3

Numerical simulations of hydromechanical
chemical coupling behavior

As observed in the previous tests, the HMC coupling behavior of rocks needs to consider elastic modulus degradation, creep strain, permeability evolution, and so on. Therefore, two kinds of damage, mechanical damage due to microcracks in solid matrix and chemical damage induced by the increase of porosity due to dissolution of matrix minerals, are defined in this numerical simulations.

4.3.1 General framework The principal phenomena to be taken into account are elastic deformation defined by the elastic strain tensor εe , plastic deformation defined by the instantaneous plastic strain tensor εp , and creep plastic strain tensor εvp , mechanical damage defined by the internal variable dm ðdm A½0; 1Þ and chemical damage defined by the internal

Stress-induced permeability evolutions and erosion damage of porous rocks

77

variable dc ðdc A½0; 1Þ. Using the assumption of small strain and displacement, the total strain increment dε is composed by an elastic part dεe , an time-independent instantaneous plastic part dεp and a time-dependent creep part dεvp . Hence, the strain increment partition rule is written as following: dε 5 dεe 1 dεp 1 dεvp

(4.5)

We assume isothermal conditions throughout this study, and the thermodynamic potential function can be expressed as follows: 1 ψðεe ; εp ;εvp ; dm ; dc Þ 5 ðε 2 εp 2 εvp Þ:ℂðdm ; dc Þ:ðε 2 εp 2 εvp Þ 2

(4.6)

1ψp ðεp ; dm ; dc Þ 1 ψvp ðεvp ; dm ; dc Þ The fourth-order tensor ℂ is the effective elastic stiffness tensor of damaged material related to the mechanical damage dm and chemical damage dc ; the terms ψp and ψvp represent the locked instantaneous plastic energy and viscoplastic energy, respectively. The standard derivation of the thermodynamic potential yields the state equation as follows: σ5

@ψ 5 ℂðdm ; dc Þ:ðε 2 εp 2 εvp Þ @εe

(4.7)

In the case of isotropic materials, the effective stiffness tensor of damaged material reads in the general form: ℂðdm ; dc Þ 5 2μðdm ; dc ÞK 1 3kðdm ; dc ÞJ

(4.8)

where kðdm ; dc Þ is the effective bulk modulus of the damaged material and μðdm ; dc Þ represents the effective shear modulus. The two isotropic symmetric fourth-order tensors are defined by J5

1 δδ 3

and

K5I2J

(4.9)

where δ denotes the second-order unit tensor, and I 5 δδ is the symmetric fourthorder unit tensor: Iijkl 5 1=2ðδik δjl
1 δilδjk Þ. Note that for any second-order tensor E, J:E 5 1=3ðtrEÞδ and K:E 5 E 2 1=3 ðtrEÞδ, which are respectively the isotropic and deviatoric parts of E. The rate form of the constitutive Eq. (4.7) can be easily written as follows: σ_ 5 ℂðdm ; dc Þ:ðε_ 2 ε_ p 2 ε_ vp Þ 1 1

@ℂðdm ; dc Þ :ðε 2 εp 2 εvp Þd_m @dm

@ℂðdm ; dc Þ :ðε 2 εp 2 εvp Þd_c @dc

(4.10)

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Porous Rock Failure Mechanics

the dot denotes time derivative of variables (or incremental variation of variables in numerical computing procedure). In order to solve the above equation, it needs the evolution laws to determine the increment of instantaneous plastic strain ε_ p , creep plastic strain ε_ vp , mechanical damage d_m , and chemical damage d_c , which will be formulated in the following sections. In addition, the dissolution of mineral in solid skeleton, diffusion, and convection of ions in pore network must verify the general mass balance equation as follows: X @ðφci Þ 5 r  ½Dðdm ; φÞrðφci Þ 1 v  rðφci Þ 1 φri @t i51

(4.11)

where φ is the porosity, D denotes the diffusion coefficient and is dependent on mechanical damage and porosity, v is the solution velocity in pore network and can be characterized by Darcy’s law, ri represents the number of dissolute moles of speci i per unit time per unit volume of fluid and can be defined by the dissolution kinetics of matrix minerals, ci is the molar density of speci i. The three terms on the right hand side are, respectively, the contribution of ion diffusion, ion convention, and the source of ion. Generally speaking, the mechanism of mass transfer is controlled by a dimensionless parameter, called Pe´clet number: Pe 5

vL D

(4.12)

where v is the scalar value of solution velocity. L is the packing length and assumed as a constant for a given porous material. Therefore, the Pe´clet number has a positive correlation with the solution velocity. According to previous investigations (Kang et al., 2003), the mass transfer is controlled by the diffusion when solution velocity is low in low porous material (Pe , 1025 ). As solution velocity increases (1022 , Pe , 10), the mass transfer is controlled by both convection and diffusion. When solution velocity is relatively high in high porous material (Pe . 10), the convection is dominated comparing with diffusion. In order to take into account the influence of pore pressure on mechanical behavior of sandstone, the effective stress concept is adopted and written as follows: σ0 ij 5 σij 1 bðdm ; dc Þp

(13)

where p is the pore pressure. bðdm ; dc Þ denotes the effective stress coefficient, which is dependent on the mechanical damage and chemical damage. We use the effective stress tensor to replace the total stress tensor and pore pressure in mechanical modeling in order to take into account the effect of pore pressure. Therefore, the mechanical behavior of saturated porous medium subjected to the effective stress is equivalent to that of dry material subjected to total stress.

Stress-induced permeability evolutions and erosion damage of porous rocks

79

4.3.2 Special model for sandstone According to the experimental results summarized in the previous section, a special model is proposed to describe the HMC coupling behavior of sandstone in this section.

4.3.2.1 Mechanical modeling According to the laboratory data of triaxial compression tests (Hu et al., 2010a), the particular forms of yield surface are defined as follows:
 f p σij ; γ p ; dm ; dc 5 Q 2 αp ðγ p ; dm ; dc ÞðCs 1 PÞ # 0 p52

σkk ; 3

q5

pffiffiffiffiffiffiffi 3J2 ;

J2 5

1 sij sij ; 2

sij 5 σij 2

(4.14) σkk δij 3

(4.15)

where P and Q are, respectively, the mean stress (compressive mean stress is taken as positive) and deviatoric stress. The parameter Cs represents the coefficient of material cohesion. Based on the laboratory data, the following plastic hardening function is proposed as follows: αp ðγ p ; dm ; dc Þ 5 ð1 2 dm Þ

 
 γp 1 αp0 1 αpm 2 αp0 p ð1 1 a1 dc Þ b 1 γp

(4.16)

where a1 is model parameters which are used to describe the effect of chemical damage on plastic response. αpm and αp0 denote, respectively, the initial value at the initial yield threshold and ultimate value at the failure point of plastic hardening variable. The parameter bp controls the evolution rate of plastic hardening. The internal hardening variable γ p is taken as the effective plastic strain that is defined by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p 2 vp vp dε dε 1 dε dε dγ 5 3 ij ij 3 ij ij p

(4.17)

where dεpij and dεvp ij are the instantaneous plastic strain and creep plastic strains, respectively. Based on these evidences and inspired by the plastic model proposed by Pietruszczak et al. (1988), the following plastic potential is used: gp 5 Q 2 ð α p 2 β p Þ ð P 1 C s Þ

(4.18)

The parameter β p defines the transition point from the compressibility zone ðαp , β p Þ to dilatation zone ðαp . β p Þ.

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Porous Rock Failure Mechanics

The viscoplastic yield surface is considered as the delayed plastic yield surface. The evolutions of instantaneous plastic yield surface and viscoplastic yield surface are related to the same internal variable, whereas the kinetics of evolution is different for each mechanism. The viscoplastic loading surface is given by
 f vp σij ; γp ; dm ; dc 5 Q 2 αvp ðγp ; dm ; dc ÞðCs 1 PÞ # 0 αvp ðγ p ; dm ; dc Þ 5 ð1 2 dm Þ

 
vp 1 γp vp  αvp 1 α 2 α 0 m ð1 1 a1 dc Þ 0 bvp 1 γ p

(4.19) (4.20)

where the parameter bvp , which is similar to the parameter bp in Eq. (4.16), controls the kinetics of viscoplastic hardening αvp , which varies from its initial value αvp 0 to vp vp maximum value αvp m . For the sake of simplicity, the two parameters α0 and αm are set to identical to the parameters αpm and αp0 in the instantaneous plastic hardening function. According to the unified viscoplastic theory, the viscoplastic loading surface should be smaller than the instantaneous loading surface, we should have αvp # αp . Therefore, the value of parameter bvp should meet the condition: bvp $ bp . For the viscoplastic potential, the same function as that for the instantaneous plastic potential is used as following: gvp 5 Q 2 ðαvp 2 β p ÞðP 1 Cs Þ

(4.21)

Based on the overstress concept by Perzyna, the viscoplastic flow rule is then determined as following:

f vp ε_ 5 γ ðT Þ Cs vp

n

@gvp @σij

(4.22)

where hxi 5 ðx 1 jxjÞ=2 is the Macauley bracket. The fluidity coefficient γ is generally dependent on temperature. Based on the classic viscoplastic theory, the following function is used:   Z γ ðT Þ 5 γ 0 exp 2 RT

(4.23)

where γ 0 is the value of fluidity at a reference temperature, Z the activation energy, T the absolute temperature, and R is the universal constant of perfect gas that takes the value of R 5 8:3144 kJ=mol=K. For the studied sandstone at room temperature, the values of parameters γ0 and Z are used as γ 0 5 500 s21 and Z 5 63; 000 N m=mol. We consider that the effective plastic strain, which is defined in Eq. (4.17), is the driving force for damage evolution. Inspired by the previous damage criteria (Hu et al., 2010b), the following exponential function is proposed: dm 5 dmc ð1 2 expð 2bd γ p ÞÞ

(4.24)

Stress-induced permeability evolutions and erosion damage of porous rocks

81

where dmc is the critical value of mechanical damage, bd is a model parameter which controls the evolution rate of mechanical damage. For the isotropic material and under the assumption that the effect of mechanical damage on the bulk modulus and shear modulus of damaged material are identical, the following relation is used to describe the effect of mechanical damage on the effective elastic modulus: ℂðdm ; dc Þ 5 ð1 2 a2 dm Þℂðdc Þ

(4.25)

where ℂðdc Þ denotes the elastic modulus of material without mechanical damage at a given chemical damage state. The coefficient a2 characterizes the mechanical damage effect on the elastic modulus and its value is related to the elastic properties of solid matrix. However, as a macroscopic modeling is concerned here, the value of a2 should be determined from relevant experimental data with loading
unloading cycle. For the sake of simplicity, we will take a2 5 1 in this study.

4.3.2.2 Mass-transfer modeling For the sandstone in our work, its permeability and diffusion coefficient are respectively 10216 m2 and 10210 m2/s, the value of Pe´clet number is determined about 102 under a moderate hydraulic gradient. Therefore, the convection is predominating in mass transfer and the mass balance Eq. (4.11) can be rewritten as follows: X @ðφci Þ 5 v  rðφci Þ 1 φri @t i51

(4.26)

To solve the above equation, we need to determine the solution velocity and reaction rate. The solution velocity can be calculated by Darcy’s law and written by v 5 K ðdm ; φÞrp

(4.27)

where Kðdm ; φÞ is the intrinsic permeability and dependent on the mechanical damage and porosity. The Carman
Kozeny equation is used to describe the relationship between the permeability and porosity, and written as following:


2  3 φ0 K ðφÞ 5 ð1 1 a3 dm ÞK0 ð12φÞ2 φ 12φ0

(4.28)

where the coefficient a3 is used to characterize the effect of mechanical damage on the permeability and may be determined by the micromechanical analysis based on the density and distribution of microcracks.

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Porous Rock Failure Mechanics

The reaction rate law is used to describe the evolution of dissolution mole number of each mineral species and dependent on the properties of solution and mineral. According to previous works (Lasaga, 1998), for constant temperature, the reaction rate ri of mineral specie i can be written as follows:  1 n m dmi i 5 Ai k i a H ri 5 fi ðΔGÞ (4.29) i dt mi0 In the above rate law, Ai is the available mineral surface area per fluid packet with units of m2 =L. ki is the reaction rate constant for a given temperature with units of mol=m2 =s, mi0 and mi denote initial and actual mineral concentration in 1 solid skeleton, aH is the term describing the pH dependence of the rate. 2 nu k 5 k25 exp4 2

0

13

2

Eanu R

0 EaH

@ 1 2 1 A5 1 kH exp4 2 @ 1 2 1 25 T 298:15 298:15 R T 2 0 13 EOH 1 1 A5
H1 nOH OH a 1k25 exp4 2 a @ 2 T 298:15 R

13
 A5 aH1 nH

(4.30) f ðΔGÞ is the function describing the effect of solution saturation state (Lasaga, 1998), which is calculated by the change of Gibbs energy. According to the mineral composition of studied sandstone, only the three main compositions quartz, Kfeldspar, and calcite are considered in this study. In addition, it is noted that the dissolution may be affected by the stress of the solid skeleton according to the pressure solution theory. However, the appropriate mathematic description of this influence is an open problem. For the sake of simplicity, the influence of stress on the dissolution rate is not taken into account.

4.3.2.3 Porosity evolution and chemical damage For most underground engineering, the mineral dissolution is usually crucial for the analysis of structure safety. Therefore, only the mineral dissolution is considered in this study. As ri is defined as the number of dissolute moles of speci i per unit time per unit volume of fluid, multiplying by both total porosity (φ) and molar volume (Miv ), yields the volumetric rate of change due to the dissolution of speci i per unit volume of bulk matrix, dφi 5 φ  MivU ri (4.31) dt and the total porosity change is written as following: dφ X dφi 5 dt dt i

(4.32)

Stress-induced permeability evolutions and erosion damage of porous rocks

83

From the mechanical view point, the increase of porosity due to mineral dissolution can be considered as a damage process. Therefore, the chemical damage variable is defined as following: dc 5

φ 2 φ0 1 2 φ0

(4.33)

When the studied material is sound, the dissolution mole number of all mineral species is zero; the chemical damage is consequently equal to zero. Although as all mineral is completely dissolute, the matrix porosity approaches unite, the chemical damage also reaches its maximum value 1. For the isotropic material and under the assumption that the effect of mechanical damage on the bulk modulus and shear modulus of damaged material are identical, the effective elastic properties of chemical damaged material are expressed as follows: ℂ ð dm ; dc Þ 5 ð 1 2 d m Þ

1 ℂ0 ð 1 1 a4 dc Þ

(4.34)

where coefficient a4 is used to characterize the mechanical damage effect on the elastic modulus, and its value can be determined from relevant experimental data showing elastic degradation with chemical degradation.

4.3.2.4 Poromechanical modeling Based on the previous study from micromechanical analysis (Shao, 1997), the following relation for effective stress coefficient is used in this study: bð dm ; dc Þ 5 1 2

K ð dm ; dc Þ 3Ks

(4.35)

where Ks is the compressibility modulus of the solid grains, which is assumed as constant. Kðdm ; dc Þ denotes the effective bulk modulus of material at a given mechanical and chemical damaged state.

4.3.3 Numerical application The simulation results of the HMC coupling behavior of sandstone are presented in this section. A numerical algorithm is set up on the basis of the proposed model and validated from decoupled tests to fully coupled tests to progressively determine the model parameters.

4.3.3.1 Simulation of chemical dissolution process The proposed model is first applied to simulate the chemical dissolution tests of the moderated porosity sandstone (Cui et al., 2008). In such tests, the rock samples

84

Porous Rock Failure Mechanics

Table 4.1 Parameters for calculating kinetic rate constants of minerals Parameters

Quartz

K-feldspar

Calcite

A (m2/g) m0 ðmol=m3 Þ M v ðm3 =molÞ

9.1 20.36 2.269 3 1022

9.1 2.55 1.086 3 1021

9.1 2.05 3.693 3 1022

1.02 3 10214 87.7 1.02 3 10214 87.7 2 21.02 3 10211 87.7 2 0.6

3.89 3 10213 38 8.7 3 10211 51.7 0.5 6.31 3 10212 94.1 2 0.82

1.6 3 1029 41.87 1 51.87 1 1 3 1023 51.87 1

Neutral mechanism Acid mechanism

Base mechanism

nu ðmol=m2 =sÞ k25 nu Ea ðkJ=molÞ H k25 ðmol=m2 =sÞ EaH ðkJ=molÞ nH OH k25 ðmol=m2 =sÞ OH Ea ðkJ=molÞ nOH

were put into the core holder with the seepage of three types of chemical solutions, including distilled water (pH 5 7), 0.01 mol/L HCl solution (pH 5 2), and 0.01 mol/L NaOH solution (pH 5 12). The porosity of rock samples was measured during the dissolution process. Based on the geochemical database of EQ3/6 (Wolery, 1992), the geochemical parameters of quartz, K-feldspar, and calcite at 25 C are obtained and presented in Table 4.1. Using the above parameters, the dissolution process of sandstone in the three solutions is simulated. The numerical results of the porosity evolution of sandstone in the three solutions are presented in Fig. 4.13. The proposed model can well describe the dependence of porosity evolution on pH. Fig. 4.14 shows the numerical results of molar density evolution of three main minerals. It is clear that the reaction rates of the three main minerals are dependent on pH. The porosity evolution is mainly attributed to the dissolution of calcite in distilled water with pH 5 7. For 0.01 mol/L HCl solution (pH 5 2), the dissolution of calcite in a great rate plays an important role in the early stage, with the expiration of calcite, the porosity evolution is gradually controlled by the dissolution of K-feldspar. In 0.01 mol/L NaOH solution (pH 5 12), both the molar density of calcite and quartz decrease simultaneously. However, because the molar density of quartz is much greater than calcite, therefore, the porosity decrease is mainly dependent on the dissolution of quartz.

4.3.3.2 Simulation of mechanical behavior The proposed model is applied to describe the mechanical behavior of sandstone. The initial elastic constants can be identified from the loading
unloading cycles performed during the stage of elastic deformation. From the triaxial compression

Stress-induced permeability evolutions and erosion damage of porous rocks 0.18

85

0.25

0.16

0.14

Data Simulation Distilled water

0.12

Porosity

Porosity

0.2 Data Simulation

0.15 Solution of pH = 2 HCl

0.1

0.1 0

20

40

60

80

0

100

20

40

60

80

Time (day)

Time (day)

(A) Distilled water with pH = 7

(B) 0.01mol/l HCI solution with pH = 2

100

0.3

Porosity

0.25 0.2

Data Simulation

0.15

Solution of pH = 12 NaOH

0.1 0

20

40

60

80

100

Time (day) (C) 0.01mol/l NaOH solution with pH = 12

Figure 4.13 Simulation of porosity evolution in solutions with different pH. (A) Distilled water with pH 5 7, (B) 0.01 mol/l HCl solution with pH 5 2, and (C) 0.01 mol/l NaOH solution with pH 5 12.

tests of sandstone (Cui et al., 2008), the average of the initial Young’s modulus and Poisson’s ratio are E0 5 9000 MPa and v0 5 0:2. The values of model parameters are given in Table 4.2. The simulation results of triaxial compression tests are presented in Fig. 4.15. The proposed model can well describe the main mechanical behavior. Based on the simulation of mechanical behavior of sandstone without chemical damage, the proposed model is then used to describe the mechanical behavior of sandstone after dissolution in distilled water (pH 5 7), 0.01 mol/L HCl solution (pH 5 2), and 0.01 mol/L NaOH solution (pH 5 12). The values of parameter a1 and a4 are given in Table 4.2. The simulation results of triaxial compression tests with confining stress of 5 MPa on sandstone after chemical dissolution are presented in Fig. 4.16. There is a good accordance with the experimental data. The proposed model is further used to simulate the long-term mechanical behavior of sandstone. Based on the creep triaxial compression tests and short-term triaxial compression tests, the parameters bvp are identified by fitting with taking into account of the condition bvp # bp . The parameter n controls the evolution of primary creep strain, its value thus can be identified by fitting the creep strain curve. The typical values of the two parameters are given in Table 4.3.

86

Porous Rock Failure Mechanics 3

Quartz

20 pH = 7 15

pH = 2 pH = 12

10 5

Molar density (mol/l)

Molar density (mol/l)

25

K-feldspar

2.5 2 pH = 7 pH = 2 pH = 12

1.5 1 0.5 0

0 0

20

40

60

80

100

0

20

40

60

Time (day)

Time (day)

(A) Quartz

(B) K-feldspar

Molar density (mol/l)

2.5

80

100

Calcite

2 1.5 pH = 7 pH = 2 pH = 12

1 0.5 0 0

20

40

60

80

100

Time (day) (C) Calcite

Figure 4.14 Evolution of molar density of three main minerals in sandstone: (A) quartz, (B) K-feldspar, and (C) calcite. Table 4.2 Value of parameters in instantaneous mechanical modeling Parameter

Value

Mechanical damage

Plastic yield and potential function

Chemical damage

bd

dmc

Cs ðMPaÞ

αp0

αpm

bp

βp

a1

15

0.8

12

0.1

1.55

5 3 1024

1.2

3

a4 3

The comparisons between model’s predictions and test data are shown in Fig. 4.17. The proposed model seems to correctly predict the evolution of strains with time and the effect of stress deviator on creep strain.

4.3.3.3 Simulation of hydro-mechanical
chemical coupling behavior We will consider a HMC coupling problem and the triaxial creep test (confining pressure of 5 MPa, axial stress of 25 and 35 MPa) on sandstone with the injection of chemical solutions (i.e., distilled water, 0.01 mol/L HCl solution, and 0.01 mol/L NaOH solution) are simulated. Based on the experimental results of permeability

Stress-induced permeability evolutions and erosion damage of porous rocks

50

(A)

40 ε3

σ1-σ3 (MPa)

87 σ1-σ3 (MPa)

60

(B)

50 εv

ε1

ε3

30

εv

40

ε1

30 20

20

σ2 = σ3 = 5MPa 10 0 –1 –0.8 –0.6 –0.4 –0.2 0

σ2 = σ3 = 10MPa

10 0.2 0.4 0.6 0.8

1

0 –1 –0.8 –0.6 –0.4 –0.2 0

1.2

(C)

ε3

–0.8

0.2 0.4 0.6 0.8

1

1.2

Strain (%)

Strain (%) σ1-σ3 (MPa)

90 80 70 60 50 40 30 20 10 0

–0.4

εv

ε1

σ2 = σ3 = 20MPa

0

0.4

0.8

1.2

1.6

Strain (%)

Figure 4.15 Simulation of triaxial compression tests of sandstone under confining stresses of 5, 10, and 20 MPa. (A) Confining stress 5 MPa, (B) confining stress 10 MPa, and (C) confining stress 20 MPa. (A)

(B) 50

σ1-σ3 (MPa)

40

40

σ1-σ3 (MPa)

30

30 σ2 = σ3 = 5(MPa) after dissolution of distilled water

20

Data Simulation

10

0 –1 –0.8 –0.6 –0.4 –0.2 0 ε3 (%)

0.2 0.4 0.6 0.8 ε1 (%)

(C)

1

1.2

40

20

σ2 = σ3 = 5(MPa) after dissolution of fluid pH = 2

10

0 –1 –0.8 –0.6 –0.4 –0.2 0 ε3 (%)

Data Simulation

0.2 0.4 0.6 0.8 ε1 (%)

1

1.2

σ1-σ3 (MPa)

30 20 σ2 = σ3 = 5(MPa) after dissolution of fluid pH = 12

10

0 –1 –0.8 –0.6 –0.4 –0.2 0 ε3 (%)

Data Simulation

0.2 0.4 0.6 0.8 ε1 (%)

1

1.2

Figure 4.16 Simulation results of triaxial compression tests with confining stresses of 5 MPa on sample after different chemical degradations. (A) Distilled water with pH 5 7, (B) 0.01 mol/l HCl solution with pH 5 2, and (C) 0.01 mol/l NaOH solution with pH 5 12.

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Porous Rock Failure Mechanics

Table 4.3

Values of parameters in viscoplastic modeling

Parameter

bvp

n

Value

9 3 1024

6

0.60

Axial strain (%)

0.50 0.40 0.30 Data 0.20

Simulation

0.10 0.00

0

100

200

300

400

500

600

700

Time (hour)

Figure 4.17 Simulation of creep tests with two levels of deviatoric stresses.

variation at different levels of mechanical damage, the value of parameter a3 is determined by fitting method as a3 5 15. The comparisons between model predictions and test data are shown in Fig. 4.18. The proposed model can well describe the different response of creep strain with the injection of different solutions.

4.4

Conclusions and perspectives

Steady and transient permeability tests were performed respectively on the moderate porosity and low porosity sandstone. The permeability variations of both sandstones under applied triaxial stress showed the similar trend and were attributed to the competition between pore compaction-induced permeability decrease and crackinduced permeability increase. The creep tests with injection of CO2 alone and CO2
brine solution were performed on the moderate porosity sandstone, CO2
brine solution had a significant influence on the creep strain and permeability comparing to the small influence of CO2 alone. This phenomenon can be attributed to the acceleration of the CO2
brine
rock reaction by the generation of carbonic acid induced by the dissolution of CO2 into the brine. The indentation tests on samples after the CO2
brine
rock reaction were also performed and indicated that the elastic modulus decreased with an increasing reaction time. A HMC coupling model was proposed for geomaterials taking into account of the mechanical damage

Stress-induced permeability evolutions and erosion damage of porous rocks

(B)

0.70

0.70

0.60

0.60

0.50 0.40 0.30 Distilled water (pH = 7 )

0.20

Data Simulation

Axial strain (%)

Axial strain (%)

(A)

89

0.10

0.50 0.40 0.30 0.01mol/l HCl solution (pH = 2)

0.20 0.10

0.00

0.00 0

100

200

300 400 Time (hour)

(C)

500

600

700

0

100

200

300 400 500 Time (hour)

600

700

0.70

Axial strain (%)

0.60 0.50 0.40 0.30 0.01mol/l NaOH solution (pH = 12)

0.20 0.10 0.00 0

100

200

300

400

500

600

700

Time (hour)

Figure 4.18 Simulation of hydro-mechanical
chemical coupling tests with two levels of deviatoric stresses and different fluid flow. (A) Distilled water with pH 5 7, (B) 0.01 mol/l HCl solution with pH 5 2, and (C) 0.01 mol/l NaOH solution with pH 5 12.

and chemical damage. A unified viscoplastic damage model is used for mechanical characterization. The reaction kinetics could account for the dissolution mechanism of three main mineral compositions under different conditions of pH. The conduction was viewed as the dominate mechanism of mass-transfer process, and the permeability was dependent on both the mechanical damage and chemical damage. The numerical algorithm and identification of model parameters were developed to simulate the experimental data. The proposed model could well account for the HMC coupling behavior of geomaterials.

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Hydraulic fracture growth in naturally fractured rock

5

Rob Jeffrey1, Xi Zhang2 and Zuorong Chen2 1 SCT Operations Pty Ltd, Wollongong, NSW, Australia, 2CSIRO Energy, Newcastle, NSW, Australia

5.1

Introduction

5.1.1 Natural fractures and hydraulic fractures Although the subject of this book is fracture and deformation of porous rock, this chapter is a departure from that subject matter and concerns itself with hydraulic fracture growth in naturally fractured rock, an increasingly important commercial topic because of the use of hydraulic fracturing for stimulation of naturally fractured shale reservoirs. Volcanic dikes and sills are exposed in outcrops at many locations and some of these have been mapped and studied in great detai+l. For example, the dikes exposed at the surface near Ship Rock in New Mexico have been studied (Delaney et al., 1986) and used to motivate new hydraulic fracturing models (Germanovich et al., 1997). A second example is the volcanic sills in the Karoo Basin in S. Africa that have been extensively studied (Chevallier and Woodford 1999). These saucer shaped sills have been studied as examples of near-surface natural hydraulic fractures by Bunger et al. (2008). Mineback studies of hydraulic fractures (Warpinski et al., 1981; Elder, 1977; Diamond and Oyler, 1985; Steidl, 1993; Jeffrey et al., 2009a,b, 1992a,b) and outcrop exposures of dikes, sills, and veins (Pollard and Aydin, 1988; Olson and Pollard, 1991) complimented by laboratory studies of hydraulic fracture interaction with discontinuities (Lamont and Jessen, 1963; Bunger et al., 2016; Daneshy, 1974; Warpinski et al., 1982; Zhou et al., 2008) have demonstrated the importance of accounting for natural fractures in determining the path and geometry of a hydraulic fracture. The hydraulic fracture can cross, arrest, offset, and/or branch during the interaction, as illustrated in Fig. 5.1, and the outcome of the interaction affects further growth of the fracture, including pressure and width in the previously formed parts of the fracture. In addition to the main fracture channel defined by the parts of the fracture completely opened by internal pressure, natural fractures that are pressurized by fluid lost from the hydraulic fracture with a corresponding decrease in effective normal stress acting across them, are dilated as a result of this pressure increase.

Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00005-1 © 2017 Elsevier Ltd. All rights reserved.

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Porous Rock Failure Mechanics

Figure 5.1 Types of outcomes from the interaction of a hydraulic fracture (HF) with a natural fracture (NF). (A) The HF crosses the natural fracture, (B) the HF is diverted into the NF with the sketch on the left showing the HF diverted into the NF and sketch on the right showing the outcome when the diverted HF reaches one tip of the natural fracture, (C) the HF is blunted at the NF, and (D) the HF diverts into the NF and then reinitiates leaving an offset in its path. after Thiercelin et al. (1987).

The reduced effective normal stress leads to a reduction in shear strength, promoting shear slip on fractures subject to shear stress. Shear-induced dilation of the fracture provides another mechanism that enhances the fracture permeability. The permeability of these natural fractures is increased substantially as their aperture increases (Laubach et al., 1998), and this leads to additional fluid loss from the main hydraulic fracture. This pressure
permeability
fluid loss positive feedback cycle is termed nonlinear or pressure dependent leakoff (Barree and Mukherjee, 1996; Jeffrey and Settari, 1998). The design of a hydraulic fracture for stimulation of a naturally fractured reservoir involves calculating the opening or width of the fracture in order to establish when and how much proppant can be injected with the fluid. The fracture length and height must be calculated to establish the surface area of the reservoir rock that the propped fracture connects to the wellbore. When the design is directed at the use of hydraulic fracturing for preconditioning in mining, the size and shape of the fracture must be predicted. The interactions between the hydraulic fracture and a natural fracture that leads to crossing, offsetting or blunting affect the fracture width

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and extent. Nonlinear leakoff also has a strong effect on fracture extent. Thus, accurate modeling of fracture growth requires an accurate method of modeling the interaction with natural fractures. The parameters that control the interaction are known only in a statistical way since the ability to directly measure them is limited by the access provided by the well or borehole or mining tunnels. Even if all fractures were completely characterized, including each fracture into a model, would result in a computationally intensive problem beyond the capacity of available computer resources. Therefore, the interaction mechanics must be included into the models in a way that is computationally efficient but is able to make reliable predictions. Research continues on both establishing the mechanics that control the interaction (crossing, branching, offsetting, and fluid loss) and developing methods for approximating the interactions that can be used to make predictions without unreasonable computational effort. The next section reviews and describes the current state of understand of crossing interactions, postcrossing interactions, and nonlinear leakoff mechanisms that impact on hydraulic fracture growth in a naturally fractured rock mass. The rest of this chapter expands on some of these topics, with various approaches contrasted and examples of applications given.

5.2

Interaction and crossing

Predicting the outcome of the interaction between a hydraulic fracture and a natural fracture has been a long-standing goal of research. Fig. 5.1 shows four types of outcomes that may result from such an interaction (Thiercelin et al., 1987) when considered in two dimensions (2D). Mapping of fractures during mineback have documented all of these interaction types as shown in Fig. 5.2. The type of interaction that occurs will affect the hydraulic fracture growth past that point and upstream of that point because of the difference in pressure needed to push fluid through planar or nonplanar features. In addition to the interaction that occurs as the leading edge of the hydraulic fracture approaches and encounters, the natural fracture, the hydraulic fracture, and natural fracture continue to interact after crossing. The dilation of the natural fracture by fluid pressure and shear induced dilation and the enhanced fluid leakoff associated with this dilation is one important interaction that continues after crossing. Any offset generated during the crossing interaction will continue to affect fracture growth as long as it is acting as a restriction to fluid or proppant flow. A crossing without offsetting may occur with an offset developing later as the hydraulic fracture opening produces shear deformation on an angled natural fracture (Jeffrey et al., 1987). The instrumented fracture experiment described by Warpinski (1985) provided direct measurement of frictional fluid pressure and fracture width in a hydraulic fracture growing through a naturally fractured rock. The measurements indicated

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Figure 5.2 Interaction types mapped by mineback. (A) A photograph showing a hydraulic fracture containing yellow proppant crossing a mineralized fracture zone, (B) a drawing from mineback of a fracture in coal showing a T-shaped fracture, illustrating diversion of the vertical fracture into a horizontal plane, (C) a drawing from a mineback of a fracture in coal showing a vertical fracture blunted by a soft clay layer, (D) a photograph showing a hydraulic fracture containing yellow proppant offsetting as it crosses a mineral filled natural fracture. Parts (A) and (C) are from a mineback experiment conducted by CSIRO at Northparkes mine (see Jeffrey et al., 2009a for details), (B) is from Jeffrey et al. (1992a), and (C) is from Jeffrey et al. (1992b).

that the fluid pressure gradient along the fracture channel was three times higher than what would occur in a smooth parallel plate fracture channel. This result demonstrated the impact of fracture offsets and roughness on fluid pressure. Later experiments and numerical calculations confirmed this strong effect (Jeffrey et al., 2009a,b; Zhang et al., 2006) that leads to higher pressure and slower growth rates for the hydraulic fractures.

5.3

Observations and experiments

In virtually all hydraulic fracture treatments, direct observation of the fracture is not possible, and its growth is determined by indirect means such as numerical

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modeling and history matching of the injection rate and pressure. Remote monitoring, such as microseismic event mapping, tiltmeter mapping, and temperature and acoustic monitoring provide additional information about fracture initiation and growth but do not give details of the fracture shape or of the interactions occurring when fracturing a naturally fractures rock mass. Direct observation of a hydraulic fracture is only possible in laboratory experiments and by mineback and mapping of full size fractures. Most mineback mapping relies on placing the fractures in an active mine to make possible cost-effective excavation. Mapping of outcrops that expose natural hydraulic fractures, such as veins, sills, and dikes, provides another source of useful data on hydraulic fracture growth in the subsurface. Examples of each of these types of direct mapping data sources are given next.

5.3.1 Mineback studies Mineback studies can occur as part of full-size experiments, designed to study specific aspects of hydraulic fracturing, or they can occur as ad-hoc observations of fractures placed for commercial reasons, most commonly for stimulation of coalbed methane wells or for gas drainage of coal ahead of mining. Mineback of hydraulic fractures placed into coal seams or coal sequences are the most numerous because of the use of hydraulic fracturing in coalbed methane well stimulation occurring near active underground mines. The mineback experiments that were carried out in ash fall and welded tuffs located on the Nevada Test Site, Nevada (Warpinski et al., 1981) were undertaken as experiments to obtain data on hydraulic fracture growth. A full-scale instrumented hydraulic fracture experiment that measured pressure and width in a hydraulic fracture was also completed (Warpinski, 1985). These studies demonstrated that the hydraulic fractures often branched or offset as they interacted with and crossed natural fractures. The instrumented fracture experiments revealed the presence of high fluid friction pressure in the fracture channel, approximately three times higher than planar fracture models predict by assuming smooth parallel plate channels. Mineback of hydraulic fractures place in naturally fractured lithified volcanic sediments in copper and gold deposits in Australia revealed similar features with both high excess pressure and branches and offsets documented (van As and Jeffrey, 2002; Jeffrey et al., 2009b, 2010; Bunger et al., 2011). In addition, two hydraulic fractures placed into andesite in Chile have been mapped by mining (Chacon et al., 2004). By modeling fracture channels with offsets, it was shown that the restriction in fracture opening at the offset was one source of the higher pressure and slower fracture growth compared to planar fracture channels (Zhang et al., 2007; Jeffrey et al., 2009b). Hydraulic fractures in coal seams have been mapped by mining since the 1970s and approximately 40 such fracture minebacks are now available in the literature. The early minebacks were motivated by the need to understand the effect of the hydraulic fractures on the stability of the roof rock in coal mines (Elder, 1977; Diamond and Oyler, 1987). These studies were undertaken by the US Bureau of Mines as part of their effort to introduce hydraulic fracturing into coal mining as a

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means of predraining gas from the coal before mining. Additional minebacks were undertaken as coalbed methane became a source of unconventional gas in the United States and Australia (Boyer et al., 1986; Steidl, 1993; Jeffrey et al., 1995). Fig. 5.3 contains a vertical section through a hydraulic fracture placed in the German Creek coal seam in the Bowen Basin, Queensland, Australia showing an overall T-shaped geometry with several additional smaller branches (Jeffrey et al., 1992a). Overall, the fractures mined consisted of one to several propped channels with offsets and branches forming at points where the main fracture interacted with bedding, shears, or natural fractures. The main features of the hydraulic fractures mapped in coal, tuff, andesite, and volcanic sediments are similar, with branching and offsetting occurring at points where the hydraulic fracture interacted with shears, bedding, or natural fractures. The strong contrasts in layer properties in the coal seam environment leads to

Figure 5.3 A vertical section through the hydraulic fracture placed into the German Creek coal seam in well ECC90, Bowen Basin, Queensland, Australia. From Jeffrey et al. (1992a).

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interactions with bedding planes (often producing T-shaped fracture geometries) that are not seen in the other mineback settings.

5.3.2 Monitoring results Fracture monitoring by indirect methods such as determining microseismic event locations and inferring fracture orientation and volume from tiltmeter data have seen wide use. More recently, both temperature and noise have been used to monitor fracturing in horizontal wells by means of fiber optic cables and distributed sensors, deployed along the wellbore. These fiber-optic based methods detect the locations of fluid injection and production along the wellbore produced by a hydraulic fracture. Microseismic monitoring has become common in stimulation of shale gas and shale oil reservoirs and provides a general map of the occurrence of shear events, which generate the microseismic events. Shearing occurs when the pore pressure increases in a natural fracture, reducing the effective normal stress acting across it to the point where the shear stress acting on it exceeds its shear strength. Shear-induced dilation of the natural fractures is associated with the shearing and enhancing the permeability of the fracture. Thus, microseismic events map out the volume that is undergoing this type of deformation and permeability enhancement.

5.3.3 Laboratory experiments Fracture crossing criteria have been tested against laboratory experiments to assess their predictive capacity (Blanton, 1986; Renshaw and Pollard, 1995). One experimental approach uses a machined surface as a model natural fracture (Bunger et al., 2016; Llanos et al., 2006). For samples that are rock-like material, details of the hydraulic fracture geometry are obtained by sectioning the sample after removing it from the testing equipment. A result from an experiment using a machined interface to represent the natural fracture (Bunger et al., 2016) is shown in Fig. 5.4.

Figure 5.4 Postfracturing section through a sandstone sample where the hydraulic fracture, driven by dyed fluid, crossed two inclined machined surfaces representing natural fractures.

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In contrast, a second approach to laboratory experiment involves growing hydraulic fractures through a sample containing many natural fractures to investigate the overall effect (Beugelsdijk et al., 2000). Fig. 5.5 contains a composite image for hydraulic fractures grown in blocks of coal, contrasting the effect of viscosity of the injected fluid. The low viscosity Wood’s metal invaded the cleats and fracture sets in the sample whereas the high viscosity epoxy was mostly contained to one fracture channel (Jeffrey et al., 1994). The image in Fig. 5.5 clearly shows the opening hydraulic fracture channel for the epoxy fluid, but the fracture channel created by the Wood’s metal cannot be distinguished from other fractures filled by fluid leaked off into the preexisting fracture network. Field-scale mineback experiments in coal, which include particulate proppants, reveal localization of the fracturing that leaves the proppant in a few channels and branches. Therefore, the Wood’s metal filling of the fractures in Fig. 5.5 illustrates the extent of nonlinear leakoff rather than a network of opening mode hydraulic fractures.

Figure 5.5 Coal hydraulic fracturing laboratory tests show the lower viscosity Wood’s metal invades most fractures while the higher viscosity epoxy is contained to a primary fracture channel. After Jeffrey et al. (1994).

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One set of fractures from laboratory experiments has been mapped in three dimensions (3D) by using serial sectioning (Kear et al., 2013), providing details of the fracture geometry not available by the usual method of cutting the block along one or two planes. Experiments in transparent materials such as glass (Bunger and Detournay, 2008), poly methyl methacrylate or PMMA (Bunger et al., 2013), and gelatine (Ito and Martel, 2002) include optical methods to measure fracture growth, shape, and opening width throughout the experiment. However, crossing interactions with natural fractures are difficult to model in the transparent materials because of the high tensile strength of the material, which reflects a small intrinsic flaw size, results in noncrossing outcomes. Recently, a 2D experimental method has been developed using rock materials that allow the trace of the hydraulic fracture to be continuously recorded by optical methods. The growth rate, opening width, and interaction of the fracture with a natural fracture can be obtained using this method (Jeffrey et al., 2015).

5.4

Analysis of HF
NF interaction

Aspects of the interaction of a hydraulic fracture with a natural fracture can be treated, after simplification, by analytical methods. For example, analytical fracture crossing criteria have been developed on the basis of a simplification of the problem. Testing the accuracy of these predictions against laboratory and field measurements determines if they can be used in hydraulic fracture design models to predict the outcome of interactions with natural fractures.

5.4.1 Fracture interaction criteria In this section, three methods to calculate the outcome of an interaction between a hydraulic fracture and a natural fracture are presented. Blanton (1986) developed an early and still useful method of predicting whether a hydraulic fracture would cross a natural fracture by simplifying the problem to a fracture on the natural fracture undergoing opening and shearing. The main hydraulic fracture feeding fluid into this fracture is not included in his analysis. Furthermore, Blanton assumed the shear stress along the sliding portion of the fracture varied linearly from zero at the opening front to the shear strength of the natural fracture at the sliding front. Fig. 5.6A shows the geometry considered and defines the parameters involved. Following Blanton’s (1986) notation, x is the distance measured from the fault center, such that x 5 b is the opening front and x 5 c is the slip front (see Fig. 5.6A). Blanton (1986) analytically derived the solution for the fracture-parallel normal stress σt at the position x, and this stress is given by Eq. (5.1) σt 5 2

f σ1 V ðx; b; cÞ c2b

(5.1)

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(A)

x σ1 = 10 MPa σt

y

c

b

τ

(B) 12 fault-parallel normal stress along-the-fault shear stress

10 Tensile Stress (MPa)

Blanton's solution 8

6

4

2

0 0

1

2

3

4

5

6

x (m)

Figure 5.6 (A) Configuration for a partially opened fracture generated by pressurized fluid under remote shear and compressive stresses. (B) Tensile normal stress along a natural fracture at a specific time by Blanton’s (1986) solution and numerically.

where f is the coefficient of friction of the natural fracture and the function V(x,b,c) is given by. "

# x2c2 1 x1c 2 x1c 2 ðx 1 bÞln 1 ðx 2 bÞln 1 ðc 2 bÞln V ðx; b; cÞ 5 π x1b x2b x2c The fracture-parallel normal stress distributions along the natural fracture given by Eq. (5.1) is compared to the stress numerically calculated using MineHF2D,

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developed by Zhang et al. (2007), based on the configuration in Fig. 5.6A. In the numerical model, the fluid is injection at a constant rate, and the slip is activated by pressurized fluid. At a specific time, a comparison between the stress found from Eq. (5.1) and with the numerical result is displayed in Fig. 5.6B, under the condition that b 5 3.28 m and c 5 4.54 m and f σ1 5 5 MPa. As shown in Fig. 5.6B, the numerical and analytical solutions of the fracture-parallel normal stress agree well with each other. The site for the maximum fracture-parallel normal stress is located within the sliding zone, as predicted by Blanton (1986). The criterion developed by Renshaw and Pollard (1995) for a fracture interacting with an orthogonal natural fracture was extended by Gu et al. (2012) to include nonorthogonal orientations. The Renshaw and Pollard criterion calculates the tensile stress on the far side of the interface, based on the stress at the edge of the process zone around the hydraulic fracture tip. If the tensile stress exceeds the tensile strength of the rock before the shear stress on the natural fracture exceeds its shear strength, then crossing is predicted to occur. The calculation is carried out before the fracture tip reaches the natural fracture, and no fluid or fluid pressure is involved directly. Gu et al. (2012) obtained a solution using the same assumptions but allowing the natural fracture to be oriented at an angle other than 90 degrees to the approaching fracture. The criterion of Gu et al. (2012) numerically solves for conditions that define a curve between predicted noncrossing and crossing conditions. Fig. 5.7 contains a plot of such curves for zero cohesion on the natural fracture and zero tensile strength of the rock. The 90-degree curve in this diagram corresponds to the Renshaw and Pollard (1995) criterion. Because crossing has been observed in laboratory experiments after fluid penetrates the natural fracture (Bunger et al., 2016) and because fluid injection rate and viscosity have been found experimentally to affect the outcome of the interaction (Llanos Rodriguez, 2015), a criterion that includes the effect of viscous flow of fluid in the fracture and along the natural fracture is needed. Chuprakov et al. (2014)

Figure 5.7 The criterion of Gu et al. (2012) for a range of natural fracture friction coefficients. For each curve, crossing is predicted if the point lies to the right of that curve.

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developed the openT criterion for this reason. The openT criterion is an analytical method to provide a crossing or noncrossing prediction for fracture interactions. The calculation includes the effect of the pressurized hydraulic fracture and accounts for injection rate and viscous fluid effects. OpenT is designed to provide a result much more quickly compared to using a fully numerical coupled model, such as MineHF2D. The hydraulic fracture width and pressure are obtained by using the Khristianovic, Gertsma, deKlerk (KGD) solution. An open channel along the frictional natural fracture is then imposed by applying half the calculated KGD width there. Fluid flow in this channel is modeled as fluid filling an empty fracture of constant width, which provides a pressure distribution along this open part of the natural fracture. Fluid is allowed to penetrate into parts of the natural fracture that have not opened, but that have a prescribed permeability. The shear stress carried by these portions of the fracture is found on the basis of it being equal to the shear strength, accounting for any reduction in the effective normal stress acting across the natural fracture where fluid has penetrated. Finally, the stress in the rock is calculated using a dislocation solution. The tensile minimum principal stress must exceed the tensile strength of the rock for crossing to be possible. If this is the case, then a small fracture flaw is assumed to exist at that location, and a fracture growth criterion is used to test if the flaw will propagate. If the flaw can propagate, then the hydraulic fracture is determined to have crossed the natural fracture, and it is allowed to extend from the flaw-location into the rock on the other side of the natural fracture. Fig. 5.8 contains the comparison, given by Chuprakov et al. (2014) between openT, MineHF2D and other criteria. The 2D numerical model MineHF2D is

Dimensionless stress contrast ΔΣ

Orthogonal intersection (β = 90°) 2

Crossing Arresting

1

OpenT criterion R&P criterion Blanton criterion

0 1E-6

1E-5

1E-4

1E-3

1E-2

1E-1 1E+0

2/s

Injection rate Q,m

Figure 5.8 A comparison for a hydraulic fracture crossing an orthogonal frictional natural fracture (from Chuprakov et al., 2014). The open square symbols and the cross symbols are predictions made using CSIRO’s MineHF2D. The OpenT prediction, given by the solid green line, is in good agreement with the MineHF2D predictions. OpenT and MineHF2D account for injection rate and fluid viscosity effects on the outcome of the interaction.

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described in detail in the next section. The dimensionless stress contrast used in Fig. 5.8 is given by ΔΣ 5 ðσH 2 σh Þ=σh .

5.5

Numerical modeling of HF
NF interaction

A numerical model is required to solve the coupled hydraulic fracture growth problem and the added complexity of the interaction of the hydraulic fracture with natural fractures. Model development can be motivated by the need to study and better understand one or more details of the process, such as the crossing interaction, or to provide a better model for design of hydraulic fracture stimulations.

5.5.1 Numerical methods A variety of numerical modeling approaches have been applied to solving hydraulic fracture and fracture
fracture interaction problems (Adachi et al., 2007). The most commonly used methods are based on the displacement discontinuity method to take advantage of the efficiency provided by a boundary element method in requiring only the boundary (the fractures) to be discretized (Clifton and Abou-Sayed, 1981; Siebrits and Peirce, 2002). Propagation of the fracture only requires adding an element or several elements to the fracture tip. However, when nonhomogeneous materials and boreholes or other openings are included, the computational efficiency of the displacement discontinuity method is reduced (Crouch and Starfield, 1983). The finite-element method has been applied to fracturing problems but requires meshing of the entire domain and typically requires a fine mesh near the fracture tip. Propagation of the fracture requires remeshing of the elements near the tip, which is difficult to implement in an efficient manner (Wawrzynek and Ingraffea, 1987). The extended finite-element method (XFEM) does not require meshing of the fracture or remeshing of the details of the tip region for propagation and is being developed for its potential in solving fracture and hydraulic fracture problems (Chen, 2013; Gordeliy and Peirce, 2015). Finite-element models can easily handle problems involving nonhomogeneous material properties or problems that involve openings. In addition, plasticity, poroelasticity, thermoelasticity, and pressure diffusion are readily available in commercial finite-element models, but computational requirements limit their use to research oriented problems.

5.5.1.1 MineHF2D model Considering the advantages of Boundary Element Method in treating crack growth, MineHF2D, an in-house research code developed by Zhang et al. (2007, 2009), is particularly suitable to address some mechanisms associated with hydraulic fracture nucleation, growth, arrest, and coalescence. When a hydraulic fracture propagates in a naturally fractured reservoir, it inevitably encounters geological discontinuities such as natural fractures and bedding planes. The hydraulic fracture response to these discontinuities depends on the injection rate and fluid viscosity, local stress

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states, intersection angle, material property contrasts, conductive and frictional properties of the discontinuities, and the existence of fluid lag. MineHF2D is designed to account for all of these aspects and can model, in 2D, complex fracture pathways that result in large gradients in injection pressure.

5.5.1.2 Finite-element cohesive zone model The cohesive zone finite-element method (CZ-FEM), which is based on the concept of a cohesive zone model (CZM) for fracture originally proposed by Dugdale (1960) and Barenblatt (1962), is one of the most commonly used tools to model fracture and fragmentation processes in various materials. Rather than the existence of an elastic crack tip region characterized by an infinite singular stress at the crack tip in linear elastic fracture mechanics (LEFM), the CZM assumes that ahead of the crack tip there is a fracture process zone (FPZ) developing according to a cohesive law. The CZM removes the physically unrealistic singularity in the crack tip stress field that is present in LEFM and provides an alternate, effective numerical method for quantitatively analysis of fracture behavior through explicit simulation of the fracture processes. Compared to the numerical methods based on LEFM in modeling hydraulic fracturing, the CZ-FEM effectively avoids the stress singularity and the nonlinear degeneracy problem associated with the fluid pressure singularity at the crack tip, which pose considerable challenges for numerical modeling in LEFM. In addition, the moving unknown footprint of the fracture and its encompassing boundary need to be solved by specifying an additional fracture propagation criterion in the simulation of hydraulic fracture propagation in the classic fracture mechanics method. Although in the CZ-FEM, the location of the crack tip is a direct, solution-dependent outcome, which increases the computation efficiency. The CZ-FEM is increasingly being used to simulate hydraulic fracture propagation and hydraulic fracture
natural fracture interaction, and its capability and accuracy have been demonstrated by comparing the modeling results with the analytical models (Chen et al., 2009; Chen, 2012; Dean and Schmidt, 2009; Carrier and Granet, 2012), numerical solutions (Chen et al., 2015; Lecampion et al., 2013; Dean and Schmidt, 2009; Sarris and Papanastasiou, 2012), and field data (Yao et al., 2015).

5.5.1.3 Extended finite-element model In modeling the problem of crack propagation, the standard finite-element model needs remeshing after every crack propagation step and the mesh has to conform exactly to the fracture geometry as the fracture propagates and thus is computationally expensive. By introducing special enriched shape functions in conjunction with extra degrees of freedom to the standard finite-element approximation within the framework of partition of unity (Melenk and Babuska, 1996), the XFEM (Belytschko and Black, 1999; Moes et al., 1999) overcomes the inherent drawbacks associated with the need to remesh propagating crack surfaces that exist when using conventional finite-element methods. By taking advantage of its numerical efficiency in modeling crack growth, the standard XFEM has been employed to investigate hydraulic fracture problems.

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Injection point

Figure 5.9 A 3-layer geometry for the simulation of a T-shaped hydraulic fracture. The shaded interfaces are discretized with cohesive zone elements.

5.5.2 Model tests for observed fracture geometries 5.5.2.1 CZ-FEM of 3-D T-shaped hydraulic fracture In many deep reservoirs, a normal fault stress regime exists so that the minimum principal stress is horizontal and hydraulic fractures grow with vertical orientation. In addition, even in a basin undergoing tectonic shortening, subject to a reverse fault stress regime, vertical hydraulic fractures are common in soft formations such as coal seams (Enever et al., 2000). Lateral growth of hydraulic fractures in coal seams requires higher injection pressures because of the interaction of the hydraulic fracture with the natural fractures in the coal and because of the restricted height growth of the vertical fracture. The increasing fluid pressure may exceed the vertical stress magnitude, which can then lead to a horizontal fracture component forming within an interface between coal and adjacent rock layers. The hydraulic fracture is then called a T-shaped fracture. Fig. 5.9 shows a 3-layer geometry used to simulate the 3D T-shaped hydraulic fracture using a CZ-FEM model. The 3-layer model consists of upper and lower layers, and a middle layer. The three layers have different mechanical properties and the upper and lower layers are typically of higher elastic modulus than the coal. As shown in Fig. 5.10, the CZ-FEM results provide detailed information on T-shaped hydraulic fracture growth such as the geometry evolution along two orthogonal planes (Chen et al., 2015). The model used to study HF
NF interaction using the CZ-FEM approach is shown in Fig. 5.11. The model couples fluid flow and rock deformation, taking into account the friction between the contacting fracture surfaces and the interaction between the HF and the NF. The extensive modeling results provide detailed.

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Figure 5.10 Geometry and injection pressure history for a 3D T-shaped hydraulic fracture predicted by the CZ-FEM. Fracture opening is indicated by the exaggerated displacement shown and by color contours with the scale in meters shown. y

σy Fluid

Rock

σx

Injection T(d) Fluid flow

Pre-defined path α

Crack

Fluid pressure Opening

Hydraulic fracture (HF)

df d0

Fracture process zone

x

Natural fracture (NF)

Figure 5.11 Schematic of a HF propagating toward a NF. The opening width of the hydraulic fracture is exaggerated to show details of the fracture opening profile and the cohesive zone.

5.5.2.2 CZ-FEM of hydraulic fracture
natural fracture interaction Quantitative information on the development of various types of HF
NF interaction behavior, and contribute toward development of an in-depth understanding of the relative roles of various parameters such as in-situ stresses, interfacial friction, intersection angle, fluid viscosity and injection rate, and initial conductivity of the natural fracture (Chen et al., 2016). Selected modeling results of the evolution HF
NF interaction and injection pressure history are shown in Fig. 5.12.

5.5.2.3 Bedding planes and parallel fractures When a hydraulic fracture intersects a bedding plane, the hydraulic fracture opening acts to promote slip deformation along the bedding plane. The shear stress

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Figure 5.12 Fracture profiles at different injection times and injection pressure history. Cases A and C are the same except that a lower coefficient of friction is used in case C. Case B is for more equal stress conditions resulting in the hydraulic fracture diverting into the natural fracture.

associated with the slip generates a tensile plane
parallel normal stress that can result in nucleation of a new fracture on the far side of the bedding plane. Depending on the slip magnitude, distribution of flaws or other structural heterogeneities along the bedding plane, more than two new fractures can be generated. During and after their nucleation, they will interact with each other. Zhang and Jeffrey (2012) found that if two fractures have different fracture lengths, caused by the sequential initiation, both can simultaneously propagate and extend to significant size because of the varying fluid pressure in both fractures. Fig. 5.13 provides a fracture geometry and pressure distributions in two such fractures under the specified conditions. The shorter fracture interacts with the longer fracture resulting in a larger pressure drop in the longer fracture at positions close to the tip of the shorter fracture. The existence of a low stress zone ahead of the shorter fracture is attributed to their continuing fracture growth, and both fractures are nucleated and propagated to a certain distance by the slip along the bedding plane. The restriction in opening along the bedding plane segments of the fractures acts as a flow divider by producing an increase in the pressure drop with flow rate.

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1.8 1.6

time = 2.683 second

10 MPa

1.4 1.2

y (m)

1 12 MPa 0.8 0.6 0.4

E = 50 GPa ν = 0.22 Kc = 1.0 MPam0.5

5 MPa

0.2 0 –0.2 –1

–0.5

0 x (m)

0.5

1

Figure 5.13 Fracture trajectory (red square symbols) and fluid pressure (indicated by the length of the blue bars plotted perpendicular to the fractures and bedding plane) along two newly created fractures (one on each side of the intersecting point). Part of the parent hydraulic fracture and the bedding plane in the homogeneous rock at t 5 2.683 s for an initially conductive bedding plane. After Zhang and Jeffrey (2012).

Fracture network formation created by hydraulic fracture propagation can occur in naturally fractured reservoirs and is important in enhancing their permeability or hydraulic conductivity. Depending on their orientation, size, and conductivity, natural fractures can act either as a lower resistance pathway or a barrier to fracture growth. To overcome a fracture flow barrier requires a pressure increase, and the pressure response varies with the ease or difficulty in fluid pathway development. Normally, for viscous fluids, a localized fluid pathway is generated by the pressurized fluid. Fig. 5.14 shows the hydraulic fracture growth in a system of preexisting natural fractures. Although hydraulic fractures can intersect many natural fracture and dilate them, a breakthrough fracture path starts at y 5 0.5 m and extends to points 3, 6, and 7 to define a localized fracture path through the natural fracture system.

5.5.3 Design modeling The hydraulic fracture design process for stimulation of horizontal wells must account for variation in reservoir characteristics (e.g., natural fracture size, orientation, spacing, and permeability) in the reservoir rock along the wellbore path. Other important

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111

1.5 4 1

y (m)

7 3

0.5

6 0 2

–0.5

–1 –1

8

5

1

–0.5

0

0.5 1 x (m)

1.5

2

Figure 5.14 Fracture pathways at the time of breakthrough. To distinguish the fracture trajectory, the red solid squares are used for natural fractures existing prior to fluid injection and the blue circles show the path for newly created hydraulic fractures. Four natural fractures are connected to the well indicated by the vertical axis. The injection rate into each individual natural fracture changes with its conductivity, and their sum is equal to the injection rate. The numbers indicate the order of intersection in time, with number 8 intersection occurring after flow breakthrough. The fracture system is subject to the applied N N stresses σN xx 5 6 MPa, σ yy 5 4 MPa, and σ xy 5 0. After Zhang and Jeffrey (2014).

parameters that affect the stimulation, such as in-situ stress, rock properties, and rock layering, also vary with location and need to be determined in the rock around the wellbore over its horizontal length. Geophysical logs, run along the wellbore, are typically the source for these data that then must be extrapolated deeper into the surrounding rock. A statistical approach to the characterization problem is required. Dimensional analysis has proved valuable by providing dimensionless groups of parameters that govern the mechanics of the problem (de Pater et al., 1994; Bunger et al., 2005; Detournay, 2004; Bunger et al., 2012). The challenge for hydraulic fracture design models is for them to incorporate the key features of the hydraulic fracture process to produce efficient and accurate predictions of fracture growth, proppant placement, and permeability enhancement of the naturally fractured reservoir.

5.6

Future directions

There is a need to continue improving the accuracy of hydraulic fracturing models, with verification of existing and new models against measurements. 3D hydraulic

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fracturing models that include the capability to model contact and friction, nonplanar growth and fracture segmentation, and interaction with natural fractures and bedding planes leading to blunting, offsetting, reinitiation, and branching should be developed. No doubt a number of different models will be developed, each with a subset of these capabilities with the models used for researching certain aspects of the larger problem. 3D but planar hydraulic fracture models have been available and used since the 1980s (Clifton and Abou-Sayed, 1981; Carter et al., 2000; Adachi, 2010) but were not widely used for design or back analysis of fracture treatments because of the computational effort they require. 3D models that allow for out-of-plane growth are available (Rungamornrat et al., 2005; Castonguay et al., 2013; Xu and Wong, 2013; Peirce and Bunger, 2013). 3D models that allow for interactions with natural fractures and hydraulic fracture branching (Weng et al., 2014) and segmentation are being developed (Nikolskiy et al., 2015) and may prove to be most useful in research applications. Coupling the hydraulic fracture to the reservoir has a long history and is being further developed to stimulation of naturally fractured reservoirs. Recent efforts have added coupling to a geomechanics model to better account for the effects of changing pressure (from both injection and production) on permeability, natural fracture deformation, and interaction with previously placed hydraulic fractures (Ji, 2007; Weng et al., 2014).

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Bunger, A., Detournay, E., 2008. Experimental Validation of the Tip Asymptotics for a Fluid-Driven Crack. J. Mech. Phys. Solids. 56 (11), 3101
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328, January. Carter, B.J., Desroches, J., Ingraffea, A.R., Wawrzynek, P.A., 2000. Simulating fully 3D hydraulic fracturing. Model. Geomechan. 525
557. Castonguay, S.T., Mear, M.E., Dean, R.H., Schmidt, J.H., 2013. Predictions of the Growth of Multiple Interacting Hydraulic Fractures in Three Dimensions, 12. SPE, New Orleans. Chacon, E., V. Barrera, R. Jeffrey, and A. van As. “Hydraulic Fracturing Used to Precondition Ore and Reduce Fragment Size for Block Caving.” In MassMin 2004, 429
534. Santiago, 2004. Chen, Z. “An ABAQUS implementation of the XFEM for hydraulic fracture problems.” In ISRM International Conference for Effective and Sustainable Hydraulic Fracturing. International Society for Rock Mechanics, 2013. Chen, Z.R., 2012. Finite element modelling of viscosity-dominated hydraulic fractures. J. Petrol. Sci. Eng. 88
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144. Chen, Z.R., Bunger, A.P., Zhang, X., Jeffrey, R.G., 2009. Cohesive zone finite elementbased modelling of hydraulic fractures. Acta Mech. Solida Sinica. 22 (5), 443
452. Chen, Z.R., Jeffrey, R.G., Zhang, X., 2015. Numerical modeling of three-dimensional T-shaped hydraulic fractures in coal seams using a cohesive zone finite element model. Hydraulic Fract. J. 2 (2), 20
37. Chen, Z.R., Jeffrey, R.G., Zhang, X., 2016. Finite-element simulation of a hydraulic fracture interacting with a natural fracture. SPE J, SPE-176970-PA. Chevallier, L., Woodford, A., 1999. Morpho-Tectonics and mechanism of emplacement of the dolerite rings and sills of the Western Karoo, South Africa. South Afr. J. Geol. 102 (1), 43
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104. Elder, C.H., 1977. Effects of Hydraulic Stimulation on Coalbeds and Associated Strata. USBM RI 8260, US Dept of the Interior 20, pages. Enever, J.R., Jeffrey, R.G., Casey, D.A., 2000. The Relationship between Stress in Coal and Rock. ARMA, Seattle. Germanovich, L.N., Astakhov, D.K., Mayerhofer, M.J., Shlyapobersky, J., Ring, Lev M., 1997. Hydraulic fracture with multiple segments i. observations and model formulation. Int. J. Rock Mech. Min. Sci. 34 (3
4). Gordeliy, E., Peirce, A., 2015. Enrichment strategies and convergence properties of the xfem for hydraulic fracture problems. Comput. Methods Appl. Mech. Eng. 283, 474
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26. Ito, G., Martel, S.J., 2002. Focusing of magma in the upper mantle through dike interaction. J. Geophys. Res. 107 (B10), 2223. Jeffrey, R.G., Settari, A., 1998. An Instrumented Hydraulic Fracture Experiment in Coal, SPE Rocky Mountain Regional/Low Permeability Symposium, 8. Society of Petroleum Engineers, Denver. Jeffrey, R.G., Vandamme, L., Roegiers, J.-C., 1987. Mechanical Interactions in Branched or Subparallel Hydraulic Fractures. SPE, Denver. Jeffrey, R.G., Byrnes, R.P., Lynch, P.A., Ling, D.J., 1992a. An Analysis of Hydraulic Fracture and Mineback Data for a Treatment in the German Creek Coal Seam. SPE, Casper. Jeffrey, R.G., Enever, J.R., Phillips, R., Moelle, D., Davidson, S., 1992b, Townsville.Hydraulic Fracturing Experiments in Vertical Boreholes in the German Creek Coal Seam, 21. Jeffrey, R.G., Paterson, L. and Pinczewski, W.V. “Investigations into hydraulic fracturing and multiphase flow in coal”, APCRC Final Report, November 1994.

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Cohesive zone models Panos Papanastasiou1 and Ernestos Sarris1,2 1 University of Cyprus, Nicosia, Cyprus, 2University of Nicosia, Nicosia, Cyprus

6.1

6

Introduction

Crack initiation and propagation is encountered in many engineering disciplines that involves advanced loading stages of mechanical behavior of solids and structures that may lead to failure. There are also a number of engineering applications in Geomechanics, not related to safety of structures, where fractures are induced or naturally occurred. The most notable example is hydraulic fracturing, a technique widely used in the petroleum industry to enhance the recovery of hydrocarbons from underground reservoirs. Some other applications in geomechanics involving fracturing include the magma-driven fractures (Spence and Turcotte, 1985), the preconditioning of rock masses in mining operations to promote caving (Jeffrey et al., 2001), the formation of barriers to stop contaminant transport in environmental projects (Murdoch and Slack, 2002), the reinjection of drilling cuttings (Moschovidis et al., 2000), the heat production from geothermal reservoirs (Legarth et al., 2005), and, more recently, the risk of induced fractures in CO2 deep geological storage (Papanastasiou et al., 2016). Modeling the initiation and propagation of fractures in continuum solids with stress analysis requires the use of a criterion that will dictate if the fracture tip is at the state to advance further. There is number of fracture propagation criteria provided by the theory of linear elastic fracture mechanics (LEFM), originally developed for metals. The criterion for fracture propagation is usually given either by conventional energy approach which states that a fracture propagates when the energy release rate reaches a critical value related to material fracture toughness or by the stress intensity approach which states that a fracture propagates when the stress intensity factor at the tip exceeds the rock toughness. The energy release rate and stress intensity approaches are essentially equivalent and uniquely related for linear elastic materials. The conventional fracture models have led to a physically meaningless singular stress field near the fracture tip, which are convenient for deriving analytical solutions of good accuracy outside the singular area. The LEFM theory has limitations to describe fracture phenomena in quasi-brittle materials such as concrete, rock, and other geomaterials when the inelastic deformation is not constrained in a small area near the crack tip. In many problems mentioned above, the presence of high confining earth stresses in combination with deviatoric loading, imposed by fractures, result in significant inelastic deformation spread around the tip. For such applications, the most robust criterion for nonlinear Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00006-3 © 2017 Elsevier Ltd. All rights reserved.

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fracture mechanics is based on the constitutive modeling of the cohesive zone or often called process zone. The cohesive zone is a region ahead of the crack tip that is characterized by microcracking along the crack path. The zone of microcracks surrounding a crack tip that is formed at peak load, also called the process zone, is a phenomenon observed in fracture of quasi-brittle materials. The main fracture is formed by interconnection of these microcracks (Labuz et al., 1985). The cohesive zone model implies that normal stress continues to be transferred across a discontinuity which may or may not be visible. The transferred stress is determined from the softening stress
strain relation that various brittle materials exhibit in calibrations tests. Thus, the cohesive zone model approach is contrasted with the conventional fracture mechanics that are based on infinitely sharp fracture models and results in singular stress at the crack tip. The cohesive zone approach cancels the stress singularity when the separation at the tail of the cohesive zone law reaches a critical value at which the cohesive traction vanishes. Sinclair (1996) showed that the cohesive traction
separation laws cancel the opening singularity produced by loading remotely from the fracture with the closing singularity produced by cohesive stresses on the fracture flanks near the fracture tip. Furthermore, this cancels the coupled fluid pressure singularity that is encountered in the analytic framework of the fluid driven problem at the crack-tip where the width is w 5 0. The process zone of microcracking that is formed at peak load in a structure made of a quasi-brittle material is associated with energy dissipation due to crack growth (Evans, 1976; Hillerborg et al., 1976). It is well known that the fracture energy must be finite and for linear fracture mechanics, this parameter is a material constant (Knott, 1973). To ensure the condition of constant fracture energy, the size of the process zone must eventually reach a limiting size (Bazant and Pfeiffer, 1987; Bazant and Kazemi, 1990). Some experiments report a constant process zone size (Le et al., 2014), whereas other experimental evidence shows that a limiting size is often not observed (Otsuka and Date, 2000). There is a clear consensus that the specimen size itself is an important factor in evaluating the process zone. In this study, we will assume that the process zone or cohesive zone is a related fracture mechanics material parameter characterized by the traction
separation calibration test. As we will show later in the results, the size of the generated process zone in boundary value problems is very different from the cohesive zone constitutive model that is implemented as propagation criterion. In examining the extensive literature concerning fracture mechanics, the cohesive zone modeling has attracted considerable attention as it represents a powerful and efficient technique for computational fracture studies. The early conceptual works related to the cohesive zone model, were introduced by Barenblatt (1959, 1962) who proposed the cohesive zone model to investigate perfectly brittle materials. Dugdale (1960) adopted a fracture process zone to investigate ductile materials exhibiting small-scale plasticity. Since then, a series of research work, to mention few, by Hillerborg et al. (1976), Needleman (1987), and Camacho and Ortiz (1996), have been oriented in the development of the cohesive zone concept in computational fracture mechanics. The work of Hillerborg et al. (1976) has been widely used as it is easier to implement into a finite element analysis. In this case, as the

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displacement
jump across the surface increases, the traction increases to reach a maximum value, related to the tensile strength, and then, it decays in a monotonic manner. Xu and Needleman (1994) have presented a potential-based cohesive zone model with cohesive elements that are inserted into a finite element mesh in advance and obey an exponential constitutive law in the normal direction. Another computational approach which has been proposed by Camacho and Ortiz (1996) was a stress-based cohesive zone model where a new surface is adaptively created by duplicating nodes which were previously bonded. Some limitations recognized in Xu and Needleman (1994) work, related to the imposed initial artificial compliance of the utilized cohesive zone law, were addressed by Geubelle and Baylor (1998) and Espinosa and Zavattieri (2003) who have adopted bilinear cohesive zone models by providing an adjustable initial slope in the cohesive law. In this chapter, we will present the use of cohesive zone models in hydraulic fracturing. Though the crack initiation and propagation have many similarities in quasi-brittle materials, we will not consider particular details from other fracture mechanics applications or discuss important issues such as the observed size and scale effects in concrete structures. For cohesive zone models in fracture mechanics of concrete, the reader is referred to the comprehensive book of Shah et al. (1995). In the next section, we will describe the technique of hydraulic fracturing and the necessity to implement a nonlinear material behavior that requires robust propagation criteria such as the cohesive zone model. We will present also in brief all the involved processes in hydraulic fracturing with emphasis in the cohesive zone modeling as the propagation criterion. In Section 6.3, we describe the geometry of the hydraulic fracturing model and the input parameters upon which the results are based. In Section 6.4, we present results for different parameters that characterize the cohesive zone modeling and their influence on the process zone, propagation pressures, and fracture dimensions in hydraulic fracturing. The main conclusions are summarized in the last section.

6.2

Hydraulic fracturing

The hydraulic fracturing technique involves the pumping of a viscous fluid from a well into the rock formation under high fluid pressure to fracture the reservoir. The pumping of fluid is maintained at a rate high enough for the fluid pressure to overcome the flow friction losses, the minimum in-situ stress or closure stress, the resistance to splitting the rock and hence to propagate the fracture. The initiated fracture as propagating in a complex stress field near the wellbore will reorient itself to propagate further in the direction of lease resistance which is perpendicular to the minimum in-situ compressive stress. During the pumping process, material like sand, called proppant, is gradually mixed with the fracturing fluid to ensure that the fracture will remain propped open after the pumping stops. A permeable channel of high conductivity will hence be formed for oil or gas to flow from the reservoir in the well (Economides and Nolte, 2000).

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Hydraulic fracturing modeling involves the coupling of various complex physical processes including (1) the viscous flow of the fluid in the fracture and the leak off in the formation, (2) the rock deformation of the surrounding medium induced by the fluid pressure on the fracture surfaces in the presence of high confining stresses, and (3) the rock splitting and fracture propagation. In most models, including analytical solutions and commercial design codes, the solid deformation is modeled with the elasticity theory, represented by an integral equation that determines the nonlocal relationship between the fracture width and the fluid pressure. The fluid flow is modeled by lubrication theory, expressed by a nonlinear partial differential equation that relates the local fluid flow velocity, the fracture width, and the gradient of the pressure. The fracture propagation is assumed to take place when the stress intensity factor at the tip reaches a critical value equal with the rock fracture toughness which in many cases is ignored assuming to be close to zero. In field operations, attention is focused on the prediction of the wellbore pressure which is normally measured during the treatment and is the only parameter available to evaluate or to redesign in real time the operation. Classical hydraulic fracturing simulators often underestimate the measured down-hole pressures. Research work involving surveying on net pressures (difference between the fracturing fluid pressure and the far-field confining stress) indicated that the net pressures encountered in the field are on average 50
70% higher than the predicted by conventional models. These observations have triggered a series of new ideas and dedicated studies which looked into the importance of the rock plastic deformation in hydraulic fracturing (Papanastasiou and Thiercelin, 1993; Papanastasiou, 1997; 1999a, 1999b; Van Dam et al., 2002). Sarris and Papanastasiou (2012, 2013, 2015) extended recently these studies to account for the pressure diffusion and porous behavior of the rock deformation. A common characteristic of these studies is the use of the cohesive zone law to propagate fractures in order to investigate the inelastic behavior of rocks in hydraulic fracturing. The influence of the parameters of the process zone on hydraulic fracturing results was studied by Sarris and Papanastasiou (2011) and Carrier and Sylvie (2012). Some early studies utilizing the cohesive zone model in hydraulic fracturing include the important work of Boone et al. (1986) and Boone and Ingraffea (1990) in which they used the cohesive zone approach to model the fracture process in impermeable and permeable rocks. Improving the research work in hydraulic fracturing, Schrefler et al. (2006) proposed a model, with a tip velocity provided as part of the solution algorithm and propagated hydraulic fractures in an unknown path that may enucleate everywhere depending only on the stress and pressure fields. In this contribution, we focus on the influence of the constitutive cohesive zone characteristics on the size of process zone and consequently on the obtained results in hydraulic fracturing modeling. Different types of models are used for the mechanical behavior of the cohesive zone to account for rigid
softening to elastic
softening behavior of rocks. For all models, the normal work of separation, which gives some indication of the implications of the cohesive surface characterization for fracture toughness in plane strain mode I fracture, is maintained the same. The resulting fracture shape, with the cohesive zone, influences the pressure

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profile in the fracture and the stress distribution near the tip which in turn affects the overall characteristics (fracture length, width, and propagation pressure) of a hydraulic fracture. Furthermore, we demonstrate that the existence of the confining stresses influences further the size of the process zone. The resulting process zone is an important parameter that may explain partially the unexpected high fluid pressures observed in hydraulic fracturing field operations. The models presented were developed for plain strain geometry taking into consideration the symmetry conditions. This geometry is appropriate for modeling short fractures with fracture height relatively greater than the fracture length. Furthermore, this geometry is also appropriate for examining tip effects as the deformation of any arbitrary fracture shape is approximately planar near the tip. The fracture propagates perpendicular to the minimum in-situ stress and remains planar. This predefined path for the propagation is also convenient with the cohesive zone numerical approach. For the stress deformation, we assume that rock obeys the equation of elasticity or poroelasticity. Extension to inelastic deformation can be easily implemented through the principle of effective stresses and the coupling of the pore pressure with volumetric changes (Sarris and Papanastasiou, 2012, 2013, 2015). For the sake of completeness, we describe next the involved physical processes: the fluid flow, the rock deformation, the fracture propagation, and the methodology that was adopted in the numerical model.

6.2.1 Fluid flow The physical process of the fluid driven fracture involves the pumping of a viscous fluid that pressurizes the fracture surfaces which deform. While increasing the pressurization, critical loading condition will reach ahead of the tip splitting the rock and driving hydraulically the fracture. Thus, this process reveals that there is a strong coupling between the moving fluid, rock deformation, and fracture propagation. The fluids that are used in hydraulic fracturing are normally power law with shear-thinning behavior which means that the viscosity decreases with increasing shear rate. In order to avoid this complex fluid behavior, a simple appropriate model for fluid flow in a fracture is assumed to follow the lubrication approximation. It assumes laminar flow of an incompressible uniformly viscous Newtonian fluid and accounts for the time dependent rate of crack opening. The continuity equation which imposes the conservation of mass in one dimensional flow is dq dw 2 1 ql 5 0 dx dt

(6.1)

where q is the local flow rate along the fracture in direction x, ql is the local fluid loss in rock formation, and w is the local crack opening. Eq. (6.1), which accounts for the fluid leak-off from the fracture surface into the rock formation, can be used to determine the local flow rate q.

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The second equation is derived from the conservation of momentum balance. For a flow between parallel plates, the lubrication equation, which relates the pressure gradient to the fracture width for a Newtonian fluid of viscosity μ, yields q5u  w52

w3 dp 12μ dx

(6.2)

where p denotes the fluid pressure, and u is the average velocity of the fluid over a cross section of the fracture. Eq. (6.2) determines the pressure profile along the fracture from the local width and the local flow rate. According to Eq. (6.2), the pressure gradient, and hence the solution, is very sensitive to fracture width. Therefore, the largest part of the pressure drop takes place within a small area near the tip where the width decreases significantly before it vanishes at the tip.

6.2.2 Rock deformation The basic theory of poroelasticity was initially introduced by the pioneering work of Biot (1941). Since then, many researchers have contributed to its further development. The theory originally was developed for soil mechanics especially for consolidation problems. The Biot poroelastic theory was reformulated in a more physically relevant manner to account for poroelastic effects by Rice and Cleary (1976). For the definition of a poroelastic system, five material constants are required. These material constants include the drained shear modulus G, the drained Poisson ratio ν, the undrained Poisson ratio ν u, the Skempton’s pore pressure coefficient B, and the intrinsic permeability κ. Rice and Cleary (1976) have successfully linked these constants to micromechanical parameters that can be easily obtained for any soil or rock type material. These micromechanical parameters are the porosity n, the fluid bulk modulus Kf, the solid grain bulk modulus Ks, the porous bulk modulus for the solid skeleton K, the Poisson ratio ν, and the permeability κ. The total stresses σij are related to the effective stresses σ0ij through σij 5 σ0ij 2 ap

(6.3)

The effective stresses govern the deformation and failure of the rock. The poroelastic constant a is independent of the fluid properties, and it is defined as α5

3ðvu 2 vÞ K 512 Bð1 2 2vÞð1 1 vu Þ Ks

(6.4)

An important distinction when applying this formulation to rock is to consider the compressibility of the constitutive materials. For soils, B and α are equal to unity, but in rocks, these are significantly less than one. The problem is stated here using the effective stress principle for porous media, and the solution is limited to a 2-D formulation. The theory of poroelasticity can be

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125

approximated numerically using the finite element method and a standard Galerkin formulation as described in Zienkiewicz and Taylor (1991) and Lewis and Schrefler (2000). The finite element equations in matrix notation are as follows:
 ½K fug 1 ½L p 5 fF gðStiffness equationÞ

(6.5)


: 
: 

 ½S p 1 ½LT u 1 ½H  p 5 q ðFlow equationÞ

(6.6)

where u are the nodal displacements, p are the nodal pressures, F are the nodal forces, q are the nodal flows, ½K is the stiffness matrix, ½L is the coupling matrix, ½H is the flow matrix, and ½S is the compressibility matrix. In a discretized form, the unknown field parameters u and p are expressed by the nodal values, and the interpolation functions enter in the calculation of the matrices as
 p 5 Np p ;

u 5 N u fug;

ε 5 B fug

(6.7)

where N p and N u are the nodal shape functions for pressure and displacements, respectively. B is the matrix for the strain ε that contains the derivatives of the displacement interpolation functions. The definitions of matrices in the system of Eqs. (6.5) and (6.6) in a 2-D formulation are given by the following expressions: ð ½K  5

ð

Ω

BT DBdΩ;

ð ½S 5

Ω

ðN p ÞT

1 p N dΩ; Q

½ L 5 a

Nu

8 9 d > > > > > = < dx >

p

N dΩ d > > Ω > > > > : dy ; 9Τ 0 8 1 d d > > > > > > > ð = B dx C < dx > B C ½H  5 κ N u N p B d N p CdΩ > > @ A d Ω > > > > > ; : dy > dy

(6.8)

where D is the elasticity matrix for drained material. The quantity 1=Q takes the following forms:   1 φ 12φ 12a 1 αð1 2 αBÞ 1 9ðvu 2 vÞð1 2 2vu Þ 5 5 ; 5 1 2 ; Q Kf Ks Ks Q BK Q 2GB2 κð1 2 2vÞð11vu Þ2 (6.9) It is out of the scope of this chapter to present more details on the discretization of the fluid flow in the fracture and its coupling with the rock deformation. For more details on the discretization, coupling, iterative, and continuation algorithms for the problem of hydraulic fracturing, the reader is referred to Papanastasiou (1999a,b).

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6.2.3 The cohesive zone model as fracture propagation criterion The cohesive zone model implies that normal stress continues to be transferred across a discontinuity which may or may not be visible as shown in Fig. 6.1. This stress is determined from the softening stress
strain relation that various rocks exhibit in calibrations tests. This transferred normal stress is a function of the separation and falls to zero at a critical opening and then the fracture propagates. The evolution of the crack is governed by energy balance between the work of the external loads and the sum of the bulk energy of the uncracked part and the energy dissipated in the fracture process. The main mathematical difficulty is given by the fact that the fracture energy depends on the opening of the distributed microcracks. To simplify the mathematical difficulties, it is assumed that the cohesive zone localizes, due to its softening behavior, into a narrow band ahead of the visible crack. The constitutive behavior of the cohesive zone is defined by the traction
separation relation derived from laboratory tests. The traction
separation constitutive relation for the surface is such that, with increasing separation, the traction across this cohesive surface reaches a peak value and then decreases and eventually vanishes, permitting for a complete separation. Simple cohesive zone models can be described by (A)

Stress free crack

Inelastic stress distribution

Elastic stress distribution

δIC

X

Visible crack True crack

(B) 0.6

σt

Traction σ (MPa)

0.5

Rigid-softening Elastic-softening

0.4 0.3 0.2 0.1 0 0

0.0001

0.0002

0.0003

0.0004

δIC

0.0005

Separation δ (m)

Figure 6.1 (A) Representation of the fracture process zone and (B) the constitutive cohesive zone law.

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127

two independent parameters which are usually, for mode-I plane strain, the normal work of separation or the fracture energy GIC and either the tensile strength σt or the complete separation length δIC (Papanastasiou, 1999a,b). In those models, even though the implemented cohesive constitutive equation followed a simple rigid
linear softening response, the calculated normal to the propagation direction stress distribution ahead of the open crack exhibited a smooth nonlinear response. Nevertheless, an additional parameter in these models is the slope of the initial loading which may define a range from rigid
softening to elastic
softening response under tensile stress state. In order to investigate the main characteristics of the full curve in hydraulic fracturing, Sarris and Papanastasiou (2011) carried out computations for different initial slopes to simulate a rigid
softening to elastic softening behavior. The transition from the elastic softening to the rigid
softening was carried out by increasing the initial slope of the constitutive cohesive law by five times in each model. The case of the most rigid behavior corresponds to 20 times the stiffness of the most elastic case. In all cases, the area under the curve which is related to the work of separation is maintained the same (Fig. 6.1B). The area under the traction
separation curve equals with fracture energy GIC which is the work needed to create a unit area of fully developed crack. For elastic solids, this energy is related to the rock fracture toughness KIC through (Rice, 1968; Kanninen and Popelar, 1985). 2 KIC 5

GIC E 1 2 ν2

(6.10)

where E is the young modulus, and ν is the Poisson ratio. The rock fracture toughness can be calculated from laboratory tests. For the case of the rigid
softening behavior, the traction
separation relation is uniquely determined by   δ σ 5 σt 1 2 δIC

(6.11)

where σt is the uniaxial tensile strength of the rock, and δIC is the critical opening displacement at which σ falls to zero. The value of δIC is given by the following equation (Kanninen and Popelar, 1985):   2 2KIC 1 2 ν2 δIC 5 Eσt

(6.12)

For the case of the elastic loading, the cohesive constitutive relations were augmented and modified to take into account the initial part of the curve, which is as follows:  σ 5 σt

δ δel

 (6.13)

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with the limit of elastic deformation given by δel 5

σt kn

(6.14)

where kn is the stiffness of the traction
separation relation in the loading regime with units of (MPa/m). In the postpeak softening regime, the cohesive constitutive relation is given by  ðδ 2 δel Þ σ 5 σt 1 2 ðδIC 2 δel Þ

(6.15)

In order to investigate further the influence of the cohesive zone law in hydraulic fracturing, Sarris and Papanastasiou (2011) have also studied different forms of softening behavior. For this purpose, an exponential softening was utilized that is described by (

D

Exponential

1 2 e2αððδ2δIC Þ=ðδIC 2δmax ÞÞ 5 1 2 e2α

) (6.16)

The above parameter is controlled by the exponential coefficient α. A parametric study for this variable with values of α 5 1, 2, 3, 4 was conducted. The value of the exponent α gives a measure of the curvature of the softening equation that enters in the traction
separation equation to yield   σ 5 1 2 DExponential σt

(6.17)

The critical value of the crack opening displacement is uniquely determined from Eq. (6.17). The graphical representation of the new constitutive behavior for different exponent values and the case of linear elastic
softening are shown in Fig. 6.2. The loading part of the cohesive constitutive law was kept the same (kn 5 1), and the softening part was parametrically investigated as it will be described in Section 6.4.4. For the numerical implementation of the cohesive zone model, interface elements are employed along the propagation direction. The interface elements are two-dimensional, isoparametric, 6-node, or 4-node interface element. A consistent isoparametric formulation permits modeling of curved crack surfaces and provides an element that is compatible with the 8-node quadratic displacement or 4-node linear finite elements that are used to discretize the internal domain.

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129

Linear a=1 a=2 a=3 a=4

0.5 0.4 0.3 0.2

1.10–3

9.10–4

8.10–4

7.10–4

6.10–4

5.10–4

4.10–4

3.10–4

0.0

2.10–4

0.1 1.10–4

Traction stress (MPa)

0.6

Separation displacement (m)

Figure 6.2 Linear and exponential constitutive laws for the softening branch.

6.3

Model geometry and input data

The governing equations were discretized in space with the finite element method and in time with the finite difference method. Quadratic interpolations were used for the approximation of displacement and linear for the pore pressure degrees of freedom. The sharp changes, which are expected in the geometry of the propagating tip, are dealt with placing a sufficient fine mesh around the predefined fracture path so as to ensure numerical accuracy. Depending on the required computational efficiency, a remeshing technique can be employed after some propagation step to ensure fine mesh near the tip area, by mapping the information from the old to the new mesh (Papanastasiou, 1999a,b). A hydraulic fracturing model based on a cohesive zone model is also available in the commercial software Abaqus (Hibbit, Karlsson & Sorensen Inc, 2006) and used in many studies (Sarris and Papanastasiou, 2011, 2012, 2013, 2015; Chen, 2012; Wang, 2015, 2016). Abaqus uses 4-node plain strain isoparametric elements to model the domain and 6-node cohesive elements to model the fluid flow in the fracture and the fracturing process. The cohesive elements were enhanced with two additional nodes for the modeling of the fluid flow. The example presented results in the next section that are derived from a discretized domain area of 30 3 30 m and a predefined path of the fracture at y 5 0 along which the cohesive elements were laid to save computational time. The geometry and the resulting discretization are shown in Fig. 6.3. The injection location is at the left-lower corner, and the fracture is assumed to grow in both directions along the axis-1. For a long fracture, the size of the injection wellbore is negligible and is usually ignored in the modeling. This remark, along with the condition that the injection wellbore is cased, cemented, and fully bonded with the rock formation, justifies the use of symmetry conditions (Fig. 6.3). The in-situ stresses were inserted as initial stresses and by applying the equilibrium

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30 m Plain strain elements

30 m

11 m Cohesive zone elements

3 2

1

Figure 6.3 Geometry, boundary conditions, and discretized domain.

load at the far right and top edges. At the top edge, a value of σ3 5 3.7 MPa was applied parallel to axis-3 as the minimum in-situ stress or closure stress perpendicular to the fracture plane. The fracture will propagate along the axis-1 which is parallel to the maximum in-situ stress σ1 5 14 MPa. An initial condition is also required for defining an initial fracture length for the flow. This length was considered 0.1 m, approximately equals with the perforation length from where the fracture initiates. The in-situ stresses and the initial conditions are applied in the first step to achieve system equilibrium before the propagation starts. The parameters used for the numerical computations are given in Table 6.1. These parameters include the rock properties, the pumping parameters, the in-situ stress field, and the initial conditions. The only extra parameter that is needed to consider poroelastic deformation and propagation of the poroelastic fracture is the pore pressure of the domain requiring an extra degree of freedom at the nodes of the plain strain elements. For comparison of the results, the total stress field was applied in the poroelastic analysis, whereas for the nonporous models, the corresponding effective stress field was applied. The constitutive response of the rigid
softening corresponds to a strong rock formation, whereas the elastic
softening corresponds to a softer rock formation. Four cases were considered, named as kn 5 1, 5, 10, 20 to correspond to the times that the slope of the loading branch has been multiplied. The shape of the constitutive law is shown in Fig. 6.1B where for clarity reasons only the investigated two extreme cases were plotted. The case of kn (x 5 1) corresponds to elastic
softening

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131

Table 6.1 Input parameters for the computational examples Variable

Value

Rock properties Young modulus, E (MPa) Poisson ratio, ν

16,200 0.3

Pumping parameters Viscosity, μ (kPa s) Injection rate, q (m3/s m) Domain permeability, k (m/s)

0.0001 5.00E 2 06 5.88E 2 10

In-situ stress field (effective) Maximum, σ1 (MPa) Intermediate, σ2 (MPa) Minimum, σ3 (MPa)

14 9 3.7

Initial conditions Void ratio, e Pore pressure (MPa) Initial gap (perforation) (m)

Table 6.2

0.333 1.85 0.1

Cohesive zone properties

Cohesive zone properties Uniaxial tensile strength, σt (MPa) Loading stiffness, kn (MPa/m) Fracture energy, GIC (kPa m) Permeability of cohesive zone, qb (m/s)

0.5 16,200/81,000/162,000/324,000 0.224 0/5.879E 2 10

behavior and the kn (x 5 20) to rigid
softening. The properties of the cohesive zone are summarized in Table 6.2. These properties include the uniaxial tensile strength, the fracture energy which is the area under the traction
separation curve calculated to meet an equivalent fracture toughness of 2 MPa m1/2, the permeability of the elements, and the parametrically investigated loading slope of the first branch of the cohesive constitutive law.

6.4

Results and discussion

In this section, we present results to demonstrate the importance of the cohesive zone characteristics in modeling hydraulic fracturing in both elastic and porous

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Fracture halfwidth (m)

5·10–4 4·10–4 3·10–4 2·10–4 1·10–4

0

1

2

3 4 5 6 7 8 Distance from injection point (m)

9

10

Figure 6.4 Fracture profiles of a propagating fracture for a rigid
softening cohesive model.

elastic solids. The importance and power of the utilization of the cohesive zone as propagation criterion in nonlinear fracture mechanics and in particular for hydraulic fracturing in weak formations was demonstrated in Papanastasiou and Thiercelin (1993), Papanastasiou (1997, 1999a,b), Sarris and Papanastasiou (2012, 2013, 2015), and Wang (2016). All the results presented next correspond to fractures at the state of propagation. The fractures were propagated from an initial length of 0.1 to reach 9 m long. An example is presented in Fig. 6.4 which shows the fracture profile at different propagating steps, every 1 m interval, for a fracture with cohesive zone model that follows a rigid
softening behavior.

6.4.1 The influence of the cohesive loading modulus on the process zone Fig. 6.5 shows the obtained half-width of a fracture in an elastic solid for cases of different loading slope before peak stress of the cohesive constitutive relation after the fracture reached a length of 5 m. The calculated width of the propagating elastic
softening corresponds to kn (x 5 1) and is much larger than the calculated width of the rigid
softening which corresponds to kn (x 5 20). The cusping of the fracture profiles is larger for the case of elastic
softening indicating a more ductile mode-1 behavior. The results of the rigid
softening are similar to a brittle behavior or to a pure elastic fracture without a process zone. The differences in the profile, results of Fig. 6.5, are the outcome of the cohesive zone models which were incorporated as the fracture propagation criterion. Fig. 6.6 shows the corresponding pressure profile in the fractures when the visual tip reached a distance of 5 m. The fluid front position is found to be at the point where the fluid pressure changes sign or falls to zero. It was assumed in these computations, the formation domain and the process zone are impermeable, and the pressure drop takes place mainly at the visual tip, which is at 5 m for these specific parameters.

Fracture halfwidth (m)

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133

1·10–3 9·10–4 8·10–4 7·10–4 6·10–4 5·10–4 4·10–4 3·10–4 2·10–4 1·10–4

1 x kn 5 x kn 15 x kn 20 x kn

0

1

2

3

4

5

6

7

8

Distance from injection point (m)

Figure 6.5 Fracture profiles for different values of the cohesive loading modulus.

Pressure profile (Mpa)

6

1 x kn 5 x kn 15 x kn 20 x kn

5 4 3 2 1 0

1

2

3

4

5

6

7

8

Distance from injection point (m)

Figure 6.6 Fluid pressures in the fractures for different values of the cohesive loading modulus.

The large difference, in the results of Figs. 6.5 and 6.6 between the rigid
softening and elastic
softening process zone behavior, is clearly problem dependent as for hydraulic fracturing, the size of the process zone influences the applied load of the problem. In contrast, for a fracturing problem under remote tensile loading, such a big difference in the results should not be not anticipated. Fig. 6.7 shows the net pressure (pressure in the fracture 2 remote confining stress) during fracture propagation. Much higher pressure is needed to propagate the fracture with elastic
softening cohesive behavior. The pressure decrease with fracture length is more pronounced in the case of the elastic
softening behavior compared to the pressure drop in the case of the rigid
softening behavior. However, as demonstrated in Fig. 6.7, for very long fractures the net pressure for all cases will tend to zero as the length of the process zone diminishes compared to the fracture length.

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4.0

1 x kn 5 x kn 15 x kn 20 x kn

Net-pressure (MPa)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

1

2

3

4

5

6

7

Distance from injection point (m)

Cohesive stress (MPa)

Figure 6.7 Net pressures vs fracture length for different values of the cohesive loading modulus.

1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5 4.8

1 x kn 5 x kn 15 x kn 20 x kn

5.2

5.6

6.0

6.4

6.8

7.2

7.6

8.0

Distance from visual tip (m)

Figure 6.8 Distribution of cohesive stress (normal to propagation direction) in front of the fracture for different values of the cohesive loading modulus.

Fig. 6.8 shows the distribution of the cohesive stress (stress transferred normal to the propagation direction) in front of the fracture visual tip for the different values of the loading slope of the constitutive cohesive model. The results reveal that the tensile stress is contained in a small region near the tip with its maximum value equal to the assumed tensile strength of the rock. During the fracturing process, there is a relief of the compressive stresses ahead of the fracture tip followed by a complete separation when the crack opening reaches the critical value defined in the propagation criterion. The elastic
softening model generates much longer process zone compared to the rigid
softening model. The short process zone generated by the rigid
softening model suggests that its results can be compared for validation with the results of an elastic fracture loaded with a uniform internal pressure. It seems that the loading slope of the constitutive cohesive model influences to a large

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135

extent the size of the generated process zone in hydraulic fracturing imposing larger fluid-lag area near the tip, and as a result, wider fractures are produced, and higher pressures are needed for propagating an elastic
softening cohesive fracture.

6.4.2 The influence of the confining stresses on the process zone Furthermore, we investigated the role of the confining stresses and its interaction with the cohesive zone in hydraulic fracturing. It was suggested that the existence of the confining stresses in field conditions may increase the resistance of rock to fracturing leading to an apparent increase of the fracture toughness. The calculations were repeated here for the two extreme cases, rigid
softening and elastic
softening for a stress field where the confining stresses were doubled. Fig. 6.9 shows that the fracture width profiles were almost doubled when the confining stresses were doubled for both rigid
softening and elastic
softening models. Fig. 6.10 shows the corresponding pressure profiles in the fractures for the two cohesive material models and for the different confining stresses. These results are better compared in Fig. 6.11 where the net pressures are plotted. Apparently, in a field with high confining stresses, higher net pressure is needed for creating a hydraulic fracture, and a wider fracture is created. We emphasize here that a hydraulic fracturing model based on LEFM does not predict any influence of the confining stress on the net pressure by definition of the net pressure which is measured with respect to confining stress. We emphasize again that the differences in the results of Figs. 6.9
6.11 are due to the longer process zone which is created under higher confining stresses as clearly shown in Fig. 6.12. The difference in the results is expected to diminish with increasing fracture length. Nevertheless, there are many applications where short hydraulic fractures are created in weak rock formations. For example, the propagation of a short fracture can also be used to interpret the results of the

Fracture halfwidth (m)

1.4·10–3

E-S, 2 x σ3 E-S, 1 x σ3 R-S, 2 x σ3 R-S, 1 x σ3

1.2·10–3 1.0·10–3 8.0·10–4 6.0·10–4 4.0·10–4 2.0·10–4 0

1

2

3

4

5

Distance from injection point (m)

Figure 6.9 Influence of the in-situ stress field on the fracture apertures for elastic
softening (E
S) and rigid
softening (R
S) cohesive laws. The cases marked with X2 correspond to a double value of the confining stresses.

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Pressure profile (MPa)

14

E-S, 2 x σ3 E-S, 1 x σ3 R-S, 2 x σ3 R-S, 1 x σ3

12 10 8 6 4 2 0

2

1

3

4

5

Distance from injection point (m)

Figure 6.10 Influence of the in-situ stress field on the pressure distributions for elastic
softening (E
S) and rigid
softening (R
S) cohesive laws.

Net-pressure (MPa)

8

E-S, 2 x σ3 E-S, 1 x σ3 R-S, 2 x σ3 R-S, 1 x σ3

7 6 5 4 3 2 1 0

1

2

3

4

5

Distance from injection point (m)

Cohesive stress (Mpa)

Figure 6.11 Influence of the in-situ stress field on the elastic net pressures for elastic
softening (E
S) and rigid
softening (R
S) cohesive laws. 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5

E-S, 2 x σ3 E-S, 1 x σ3 R-S, 2 x σ3 R-S, 1 x σ3

2

3

4

5

Distance from visual tip (m)

Figure 6.12 Influence of the in-situ confining stresses on the cohesive stresses in front of the fractures for elastic
softening (E
S) and rigid
softening (R
S) cohesive laws.

Cohesive zone models

137

minifrac calibration test that is carried out first in situ for determining parameters such as the formation permeability and the closure stress that are further used for modeling the long hydraulic fractures. Other applications include the creation of short fractures near the wellbore for stimulating the hydrocarbon production while avoiding sand production in weak formations.

6.4.3 The influence of the pore pressure and diffusion on the process zone

Fracture halfwidth (m)

Computations were also carried out for hydraulic fracturing in a porous material taking into account the fluid diffusion in the surrounding formation. The results were compared with those of the nonporous material where fluid diffusion and leak-off in the formation were ignored. The objective of these calculations was to investigate further how the results of the cohesive model will be affected by the existence and the changes of the pore pressure in the formation. Fig. 6.13 shows the half-width of a propagating fracture in the poroelastic solids for the cases of different loading slope of the cohesive constitutive law after the fracture reached a length of 3 m. Comparing the results of Fig. 6.5 for the nonporous fracture, with the results of Fig. 6.13 for the porous fracture, it is evident that the porous fracture profiles are wider especially for the case of the elastic
softening model, whereas for the case of the rigid
softening, the difference is negligible. The porous fractures reached about the same width at a shorter length (3 m) compared to a longer length (5 m) for the nonporous fractures. Fig. 6.14 shows the pressure profile in the fracture and process zone during propagation when the visual tip reached a distance of 3 m. The fluid-pressure drops abruptly near the visual tip. The pressure ahead of the fracture tip decreases to values below the initial formation pressure and far away from the tips tends asymptotically to its initial undisturbed value of 1.85 MPa. 1·10–3 9·10–4 8·10–4 7·10–4 6·10–4 5·10–4 4·10–4 3·10–4 2·10–4 1·10–4

Porous, 1 x kn Porous, 5 x kn Porous, 15 x kn Porous, 20 x kn

0

1

2

3

4

5

6

7

8

Distance from injection point (m)

Figure 6.13 Aperture of porous
elastic fractures for different values of the loading slope of the cohesive laws.

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Pressure profile (MPa)

10

Porous, 1 x kn Porous, 5 x kn Porous, 15 x kn Porous, 20 x kn

8 6 4 2 0

1

2

3

4

5

6

7

8

Distance from injection point (m)

Figure 6.14 Pressure profile in the porous
elastic fractures for different loading slope of the cohesive law.

Net-pressure (MPa)

8

Porous, 1 x kn Porous, 5 x kn Porous, 15 x kn Porous, 20 x kn

6 4 2

0

1

2

3

4

5

6

7

Distance from injection point (m)

Figure 6.15 Net pressures for porous
elastic fractures and for different loading slope of the cohesive law.

Fig. 6.15 shows the net pressure (pressure in the fracture 2 remote confining stress) during fracture propagation as a function of the fracture length for the porous models. In comparison with Fig. 6.7, the required net pressure that is needed for propagating the porous fracture is higher than the net pressure required for propagating the nonporous fracture, especially for short fractures. In order to estimate the size of the cohesive zone in the porous models, we have plotted in Fig. 6.16 the profile of the stress normal to the propagation direction ahead of the fracture tip. It is observed again that the process zone size is much larger in the case of the elastic
softening behavior, especially in the case of the porous model, suggesting that the pore pressure in the formation has a strong interaction with the stress concentration near the fracture tip. It is noted that, although a linear softening behavior in the cohesive model was implemented, the calculated stress profiles from the resulting separations are sufficiently nonlinear. This outlines the adequacy of implementing simple softening relations in capturing the nonlinear stress distribution ahead of the crack tip.

Cohesive stress (Mpa)

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139

1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5

Porous, 1 x kn Porous, 5 x kn Porous, 15 x kn Porous, 20 x kn

3

4

5

6

7

Distance from visual tip (m)

Figure 6.16 Distribution of stress normal to the propagation direction for porous models.

Fracture halfwidth (m)

9·10–4

Porous, linear Porous, a = 1 Porous, a = 2 Porous, a = 3 Porous, a = 4

8·10–4 6·10–4 5·10–4 3·10–4 2·10–4 0

1

2

3

4

5

Distance from injection point (m)

Figure 6.17 Width-profiles for porous elastic fractures with different form of exponential softening.

6.4.4 The influence of the softening modulus on the process zone In the next computations, we investigated the influence of the exact softening form after peak stress on the required propagation pressures and fracture dimensions of the created fractures. We considered the exponential softening forms described by Eqs. (6.16) and (6.17) and depicted in Fig. 6.2. Fig. 6.17 shows the half-width of a propagating fracture in poroelastic solids for cases of different exponential softening. It is clear that the exact form of the exponential softening does not influence to a great degree the fracture profiles as well the pressure profile in the fracture (Fig. 6.18). Fig. 6.19 shows the distribution of the cohesive stress (stress transferred normal to the propagation direction) in front of the fractures for the different values of the exponential softening in the constitutive cohesive model. The results reveal that the

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Pressure profile (mpa)

10 Porous, linear Porous, a = 1 Porous, a = 2 Porous, a = 3 Porous, a = 4

8 6 4 2

0

1

2

3

4

5

Distance from injection point (m)

Cohesive stress (Mpa)

Figure 6.18 Pressure distributions in the porous
elastic fractures for different form of exponential softening. 1.0 0.5 0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 –3.5

Porous, linear Porous, a = 1 Porous, a = 2 Porous, a = 3 Porous, a = 4

2

3

4

5

Distance from visual tip (m)

Figure 6.19 Distribution of stress normal to the propagation direction for porous models and different form of exponential softening.

form of the exponential softening changes slightly the stress profile in the process zone, but the size of the process zone remains nearly unaffected. It is clear that the exact softening form does not change the location of the peak stress which is rather defined by the loading path of the cohesive model. The size of the process zone is defined by the peak stress and the visual tip changes length, but its influence on the hydraulic fracturing results is not significantly affected.

6.5

Conclusions

We presented the use of the cohesive zone model as a fracture propagation criterion in nonlinear fracture mechanics with application in the modeling of hydraulic

Cohesive zone models

141

fracturing. The cohesive zone model is proved to be a powerful fracture propagation criterion convenient to be used in finite element analysis that enables the implementation of nonlinear material behavior, fluid-flow in the fracture, confining stresses, pore pressure and diffusion effects, and other features that can be not addressed within the framework of linear fracture mechanics. The objective of considering the above nonlinear mechanisms that requires the cohesive zone model was to explain the elevated pressures that are needed to propagate the fractures in the field and are not correctly predicted by the conventional models. Fully coupled models for the fluid flow in the fracture, the rock deformation, and the fracturing process were implemented and solved numerically with the finite element method. In order to keep the model parameters to a minimum, rigid
softening and elastic
softening cohesive relations were considered to represent the splitting process of the rock near the fracture tip. In all cases, the area under the calibration curve of traction
separation at the crack line, which is related to the strain energy released rate and is considered to be material characteristic, is kept constant. From the analysis conducted, it was found that the size of the process zone in the BVP of hydraulic fracturing is affected by the loading branch of the cohesive zone constitutive model with the elastic
softening to result in much larger process zone than the rigid
softening model. Therefore, for propagating a fracture with an elastic
softening cohesive model higher pressure is needed, and the created fracture is wider compared to the case of a rigid
softening cohesive model. These results are clearly due to the larger process zone generated by the elastic
softening model. Furthermore, we showed that the exact form of the softening branch of the cohesive model has no significant influence on the obtained results. The existence of the confining stresses increases the size of the process zone and results in wider fractures and higher propagation pressures. The changes in the pore pressure during fracturing increase further the size of the process zone which in turn increase the propagation pressures and the dimensions of the created fractures.

Acknowledgment This contribution was based on the research program ENIΣΧ/0505/31 of the Cyprus Research Promotion Foundation.

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364. Evans, A.G., 1976. On the formation of a crack tip microcrack zone. Scr. Metallurg. 10 (1), 93
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782. Jeffrey, R.G., Settari, A., Mills, K.W., Zhang, X., Detournay, E., 2001. Hydraulic fracturing to induce caving: fracture model development and comparison to field data. In: Elsworth, D., Tinucci, J.P., Heasley, K.A. (Eds.), DC Rocks: rock mechanics in the national interest: 38th U.S. Rock Mechanics Symposium; Jul 7
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260. Kanninen, M.F., Popelar, C.H., 1985. Advanced Fracture Mechanics. Oxford University Press, New York. Knott, J.F., 1973. Fundamentals of Fracture Mechanics. Butter-worths, London. Labuz, J.F., Shah, S.P., Dowding, C.H., 1985. Experimental analysis of crack propagation in granite. Int. J. Rock Mech. Min. Sci. 22 (2), 85
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Lewis, R.W., Schrefler, B.A., 2000. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Chichester, England. John Wiley and Sons. Moschovidis, Z., Steiner, R., Peterson, R., Warpinski, N., Wright, C., Chesney, E., et al., 2000. The Mounds drill-cuttings injection experiment: final results and conclusions. Proceedings of the IADC/SPE drilling conference. Society of Petroleum Engineers, Richardson [SPE 59115]. Murdoch, L.C., Slack, W.W., 2002. Forms of hydraulic fractures in shallow fine grained formations. J. Geotech. Geoenviron. Eng. 128, 479
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461. Shah, S.P., Swartz, S.E., Ouyang, C., 1995. Fracture Mechanics of Concrete: Applications of Fracture Mechanics to Concrete, Rock and Other Quasi-Brittle Naterials. John Wiley & Sons, New York. Sinclair, G.B., 1996. On the influence of cohesive stress-separation laws on elastic stress singularities. J. Elasticity. 44, 203
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Wang, H., 2015. Numerical modeling of non-planar hydraulic fracture propagation in brittle and ductile rocks using XFEM with cohesive zone method. J. Petrol. Sci. Eng. 135, 127
140. Wang, H., 2016. Poro-elasto-plastic modeling of complex hydraulic fracture propagation: simultaneous multi-fracturing and producing well interference. Acta Mech. 227 (2), 507
525. Xu, X.-P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. and Physics of Solids. 42 (9), 1397
1434. Zienkiewicz, O.C., Taylor, R.L., 1991. The Finite Element Method, Volume 2, Solid and Fluid Mechanics Dynamics and Non-Linearity. McGraw-Hill, London.

Application of discrete element approach in fractured rock masses


7

Zhihong Zhao Tsinghua University, Beijing, China

7.1

Introduction

The in-situ stresses and their redistributions due to nature or engineering perturbations may induce the failure of fractured rock masses, in terms of the reactivation of preexisting fractures (i.e., opening, closing, and shearing) and the propagation of new fractures (i.e., nucleation, growth, and coalescence of microcracks). Fractures in rock masses have a controlling influence on the mechanical behavior of rock masses as they provide planes of weakness on which further displacement can more readily occur. Fractures also often provide major conduits for groundwater flow and contaminant migration. Therefore, an accurate understanding of the coupled thermalhydrologicalmechanicalchemical (THMC) processes in fractured brittle rock masses is a critical issue in many applications, such as underground nuclear waste repository, CO2 sequestration, and enhanced geothermal system. Discrete element methods (DEMs), including discrete fracture network (DFN) models, have been widely used to simulate the coupled THMC processes, at the local scale of single fractures or at the field scale of fractured rock masses that contain a huge number of fractures. The main advantages of DEMs are explicitly depicting the fracture geometry in relative details and considering the progressive degradation of material integrity during deformation processes (Jing et al., 2013; Lisjak and Grasselli, 2014). This chapter aims to summarize the recent developments as reported by the author in a series of papers during the past half-decade, with focus on the applications of DEMs in modeling the coupled THMC processes at the scales of single fractures and fracture networks, respectively.




This chapter is prepared for the book—porous rock failure mechanics with application to hydraulic fracturing, drilling, and structural engineering.

Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00007-5 © 2017 Elsevier Ltd. All rights reserved.

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7.2

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Main features of discrete element approach

The available numerical approaches to characterize fractured rock masses are typically classified either as equivalent (deterministic or stochastic) continuum method or DEM (Jing and Hudson, 2002). The equivalent continuum model assumes that macroscopic fractured rock masses can be treated as an equivalent porous medium. The influences of the fractures are implicitly represented in the constitutive models and other associated input parameters. According to the subcontinua considered, there are three types of equivalent continuum models: (1) the single-continuum models equivalently consider the average properties of both fractures and matrix, based on the principles of standard continuum (Berkowitz et al., 1988; Doughty, 1999; Jackson et al., 2000); (2) the dual-continuum models represent the fractures as secondary porosity contributors to the original porosity of rock matrix by breaking the porous media into blocks, and the fluid and mass transfer between the fractures and matrix was modeled as the exchange flux between the two continua (Berkowitz, 2002); and (3) in triple (or multi)-continuum models, either the matrix is subdivided, or the fractures are divided into large (globally connected) and small (locally connected) fractures (Liu et al., 2003; Wu et al., 2004). Discrete element models treat the rock matrix directly as an assembly of separate blocks or particles (Lisjak and Grasselli, 2014). Fractures are usually represented by one-dimensional (1D) line (or pipe) elements in two-dimensional (2D) models, or by 2D planar elements (in shape of circle, ellipse, or polygon) in threedimensional (3D) models (Long et al., 1982; Billaus et al., 1989; de Dreuzy et al., 2000). Therefore, the DEM model is, in theory, a more realistic representation of densely fractured rock masses, which has also been the major drive for the DEM development. As the concept of bonding between discrete elements was introduced by Potyondy and Cundall (2004), the application of DEM has been extended to simulate the transition from continuum to discontinuum (i.e., fracturing). The main disadvantage of DEMs is that very detailed information of fracture geometry (i.e., position, size, shape, and orientation) and properties (i.e., stiffness, strength, and permeability) is needed but difficult to be obtained. Consequently, it demands considerable compute memory and speed. In contrast, the equivalent continuum models are more suitable for predicting the overall behavior of fractured rock masses at relatively large scales. In order to justify the equivalent continuum approach for fractured rock masses, two conditions must be satisfied (Long et al., 1982; Jing et al., 2013): (1) a representative elementary volume (REV) must exist, so that averaging techniques can be applied over the REV to derive equivalent properties and (2) the derived equivalent properties at the REV scale, or above it, must be expressed in tensor form by the principles of standard continuum. In this context, using DEM to determine the REV and the equivalent mechanical properties have been attempted by Min et al. (2004), Baghbanan and Jing (2007), and Bidgoli et al. (2013). With the development of computational power, 3D DEM was used to study the coupled TM response of a potential repository under loading of in-situ stress and heating from spent nuclear fuel, at the scales of near- or far-field, respectively (Jing and Stephansson, 2007).

Application of discrete element approach in fractured rock masses

7.3

147

Particle-based DEM for rock fractures and matrix

7.3.1 Basic features of particle-based DEM The particle-based DEM was originally developed to model the micromechanical behavior of noncohesive granular materials (Cundall and Strack, 1979), and then the cohesive (commonly called parallel) bonds between rigid particles were proposed, in order to simulate fracturing and fragmentation processes in brittle rocks. Basically, a bonded-particle model (BPM) represents the rock matrix as an assembly of rigid 2D circular disks or 3D spheres bonded in the normal and shear directions at all contacts, which possess finite normal and shear stiffness and tensile and shear strengths (Fig. 7.1; Potyondy and Cundall, 2004). Parallel bonds provide a general way to represent cements between grains, which can carry both forces and moments. There are, in addition, other three options to represent rock matrix (Fig. 7.1): contact bonds, clumps, and smooth-joint contact models. Contact bonds behave as parallel bonds,

(A)

(B)

(C)

(D)

Figure 7.1 Illustration of the four options to represent rock matrix in particle-based DEM models. (A) Parallel bonds: stiffness is contributed by both contact and bond stiffness. (B) Contact bonds: stiffness is only contributed by contact stiffness. (C) Clumped particles: a deformable body that cannot break apart. (D) Smooth joint model: a joint obeying Coulomb sliding with dilation.

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but they do not have normal or shear stiffness. A bending moment cannot be resisted by a contact bond, and thus contact bonds can be regarded as a simplified form of parallel bonds. A clump, which is a group of glued particles, behaves as a single rigid body/grain. Particles in a clump may overlap to any extent, but cannot break apart, regardless of the forces acting upon it (Cho et al., 2007). Microcracking through the interiors of mineral grains cannot be properly represented by clumps. The smooth-joint contact model can be used to represent grain boundaries in a rock matrix (Potyondy, 2010; Ivars et al., 2011; Hofmann et al., 2015). Rock fracture segments can be represented by a group of particles (in gray; Fig. 7.2A) with zero bond strength and friction coefficient, which separate the rock matrix into two parts, similar to a freshly created tensile fracture (Cundall, 1999). The infilled gouges are assumed as circular assemblies (or clusters) consisting of a number of bonded particles. Note that the bonds at boundaries between fracture surfaces and gouges are removed. The intracluster microparameters are set to the same values as that of particles representing rock matrix. In this way, the roughness of both fracture and gouge surfaces can be naturally simulated by the arrangement of particles. The basic assumptions and laws of particle-based DEM have been clearly reported by Potyondy and Cundall (2004) and Jing and Stephansson (2007), which are not repeated in this chapter. The methodology of using the BPM to simulate fracture damage during shear, including abrasion, crushing, and microcracking is presented below. When a particle representing fracture surfaces loses all its contacts with the neighbor particles, it is regarded as an abrasive gouge particle cut (or ploughed) out by the contacting asperities (Fig. 7.2B). The abrasive gouge particles may recontact with fracture surfaces during the subsequent shear process owing to normal closure or further movement. When the corresponding component of contact force exceeds either the tensile or shear strengths, the bond fails. The space between two particles bonded previously is treated as a microcrack (Fig. 7.3).

7.3.2 Single fracture damage during shear Particle-based DEM has recently been used to study the mechanical behavior of single rock fractures, and gouge production and evolution during shearing (Park and Song, 2009; Asadi et al., 2012; Zhao et al., 2012; Zhao, 2013). Following the

Figure 7.2 Particle-based DEM model of a rock fracture segment (Zhao et al., 2012). The white and gray particles represent rock matrix and fracture surfaces, respectively. The dark particles are shear-induced abrasive gouges. (A) Before shear and (B) after shear.

Application of discrete element approach in fractured rock masses

149

B A

d

Lc

2bc = (A)

+(

−

)

+

(B)

Figure 7.3 Microcracks represented by the failed bonds (Zhao et al., 2012). (A) The short gray line segments indicate shear microcracks, and the short black line segments are tensile microcracks. (B) The microcracks are represented by a pair of smooth parallel plate surfaces. (A) Microcracks in fracture surfaces and (B) Conceptual model of microcracks (Itasca, 2008).

sample genesis procedure in Potyondy and Cundall (2004), two particle mechanics models with different lengths were built up to represent synthetic fracture segments initially without or filled in by gouge particles, respectively (Fig. 7.4). In the longer fracture segment (Fig. 7.4B and C), the average disk radius was about 0.25 mm, and this was in the size range of mineral grains of typical crystalline granite. In the shorter fracture sample filled in by gouges (Fig. 7.4D and E), the radius of the gouge particle was about 0.5 mm, and the gouge particle consisted of about 730 disks with the average radius of about 0.017 mm. All of the model parameters are listed in Table 7.1. The numerical biaxial and Brazilian testes were carried out, and it is demonstrated that the microparameters can represent the properties of fresh rock fracture or gouge materials. During the numerical test of direct shear, the upper block of the fracture sample moved in the right direction at a final constant speed of 0.5 m/s under constant normal stress condition. This shearing speed was sufficiently low to ensure that the fracture sample remained under a quasi-static equilibrium state during the shear process, and it was reached in a sequence of 10 stages to avoid inertial forces within the fracture sample. The fracture surface damage, movement and breakage of gouge particles, shear stresses, and dilation/closure were monitored during the whole shear process. Based on the numerical results of the above fracture segments, fracture damage mechanisms including abrasion, crushing, and microcracking are presented below.

7.3.2.1 Fracture segments without gouge With increasing shear displacement, the number of abrasive gouge particles and (tensile and shear) microcracks generated in the fracture surfaces increased, and more tensile microcracks were generated in the zones close to the fracture surfaces (Fig. 7.5). With increasing normal stress, the number of abrasive gouge particles decreased, whereas the number of tensile and shear microcracks increased considerably. The main reason for the decreasing abrasive gouge particles was due to the

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Figure 7.4 Particle-based DEM model of rock fracture segment with or without gouges (A) Schematic view of rough rock fracture filled by gouge particles of various sizes, (B) a relative smooth fracture segment, (C) a rough saw-tooth fracture segment, (D) a gouge particle in a short fracture segment, and (E) two gouge particles in a short fracture segment. After Zhao, Z, Jing, L, Neretnieks, I., 2012. Particle mechanics model for the effects of shear on solute retardation coefficient in rock fractures. Int. J. Rock Mech. Min. Sci. 52, 92102; Zhao, Z., 2013. Gouge particle evolution in a rock fracture undergoing shear: a microscopic DEM study. Rock Mech. Rock Eng. 46, 14611479.

fact that the abrasive particles reestablish contact with fracture surface in the closing fracture, and those gouge particles were not accounted for in this simulation. In addition, the abrasive gouge particles can also be further crushed into smaller particles. This effect cannot be considered in the present model, but will be discussed later in the results of shorter fracture segments. Overall, the thickness of the damage zone increased with the increasing normal stress.

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151

Microparameters for the intact rocks of demonstration fracture samples (Zhao et al., 2012)

Table 7.1

Parameter (unit)

Value

Particle density (kg/m3) Bond contact normal stiffness (GPa) Bond contact shear stiffness (GPa) Bond contact friction coefficient (2) Bond contact normal strength, mean (MPa) Bond contact normal strength, standard deviation (MPa) Bond contact shear strength, mean (MPa) Bond contact shear strength, standard deviation (MPa)

2600 10 3.0 0.6 20 5.0 20 5.0

(A)

(B)

(C)

Figure 7.5 The abrasive particles and microcracks in the longer fracture segments after a shear displacement of about 5 mm under different normal stresses. (A) Normal stress of 1 MPa, (B) normal stress of 5 MPa, and (C) normal stress of 10 MPa. After Zhao, Z, Jing, L, Neretnieks, I., 2012. Particle mechanics model for the effects of shear on solute retardation coefficient in rock fractures. Int. J. Rock Mech. Min. Sci. 52, 92102.

7.3.2.2 Fracture segments with gouge The gouge particles can be understood as the abrasive fragments that were generated during shear or nature gouge particles preexisted in the fracture voids. Under a low constant normal stress of 0.225 MPa, the gouge particle rolled with the moving wall, and slight surface erosion (edge damage or abrasion) occurred at the bottom

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Figure 7.6 Gouge particle evolution process in a fracture segment where the upper wall moved in the right direction at a speed of 0.5 m/s. (A) Single gouge under a normal stress of 0.225 MPa, (B) single gouge under a normal stress of 1.0 MPa, (C) double gouges under a normal stress of 0.225 MPa, and (D) double gouges under a normal stress of 1.0 MPa. After Zhao, Z., 2013. Gouge particle evolution in a rock fracture undergoing shear: a microscopic DEM study. Rock Mech. Rock Eng. 46, 14611479.

of the gouge particle (Fig. 7.6). When the normal stress increased to 1.0 MPa, gouge particle breakage was observed after a shear displacement of 1.17 mm. However, the gouge particle still rolled by a small angle at the beginning period of the shear process (before breakage), accompanied by microcracking inside the gouge particle and surface erosion (Fig. 7.6). With increasing shear displacements, the microcracks fully propagated through the gouge particle and full breakage

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153

occurred. The original gouge particle was crushed into four major pieces under the high normal stress of 1.0 MPa, with many other smaller comminuted particles. Initially, the two gouge particles were in contact with each other, but they were separated during the shear process, under a low normal stress of 0.225 MPa (Fig. 7.6). During shearing, the left gouge particle lost contact with upper wall and stopped moving sometimes. In contrast, the right gouge particle was in contact with the upper wall the whole time, so it rolled much further than the left one. Under a high normal stress of 1.0 MPa, both gouge particles moved in a similar pattern (Fig. 7.6) because they were fully in contact with fracture walls under high compression. The surface erosion also became much more severe as the gouge particles must overcome stronger friction to move under the higher normal stress. Note that some surface erosion was induced by the relative friction at the contacts of both gouge particles. More small pieces were plowed off from gouge particles with increasing normal stress after the same shear displacement, but the right gouge particle was damaged more seriously than the left one. This indicates that the gouge shape and roughness of the fracture surface also influence the gouge evolution. The characteristics of gouge evolution under shear are strongly dependent on the applied normal stresses and the number of gouge particles (Fig. 7.7). When multiple gouge particles exist in a fracture segment, simultaneously, some of them may lose contact with the moving fracture wall under low normal stress. This can partly explain the persistence of survivor gouge particles in a sheared fracture filled in with a large number of gouge particles (Mair and Abe, 2008). The interaction between gouge particles can induce surface erosion (plowed pieces) at the interfaces between gouge particles. With increasing normal stress, the number of crushed gouge pieces increased significantly, induced by high contact forces between the gouge particles and fracture walls. In addition, more abrasive particles were also plowed off from the asperities of fracture walls with increasing normal stress.

Figure 7.7 The dependence of gouge evolution during shear on the normal stress and number of gouge particles.

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Under a low normal stress, the roughness of both fracture walls and gouge particle surface can result in dilation during the rolling process. When normal stresses increased, fracture aperture gently closed during the initial period. This resulted from surface erosion, but gouge particle rolling was still the dominant pattern of movement. With further shear, fracture aperture closure became more drastic due to the occurrence of gouge particle breakage, so the main type of gouge evolution changed from rolling into comminuting. The obtained modeling results can somehow explain the shear mechanisms of a natural rock fracture. For a fresh fracture undergoing shear, abrasive gouge particles can be plowed from contacting asperities, and then the shear movement may change from surface sliding to gouge rolling. Consequently, it is predicated that the measured friction angle should decrease, which is demonstrated by the shear stressshear displacement curve in Pereira and de Freitas (1993). Similarly, it was shown that rock fractures may exhibit smaller friction angles under higher normal stresses, and one possible reason should be the generated abrasive particles (without failure) that may reduce the contact area between the two fracture walls. Meanwhile, rolling is much easier than sliding. Further increasing normal stress would crush the gouge particles, so the shear stress also increases but still lower than the peak shear stress. Generally, many experiments have shown that the infilling materials can significantly decrease the shear strength (Papaliangas et al., 1993; Pereira, 1997). The gouge particles contribute to this decreasing shear strength in terms of rolling and reducing sliding surfaces.

7.3.3 Thermal-induced rock weakening and failure Thermal loading can generate two types of microcracks in rock matrix via different mechanisms: cycling and thermal gradient-induced cracks. Cycling cracks form because of a mismatch between thermal expansion coefficients of adjacent mineral grains in a homogeneous temperature field, whereas thermal gradient-induced cracks result from thermal stresses induced by temperature gradients exceeding the local grain strength (Jansen et al., 1993). In this section, particle-based DEM method is used to simulate the process of thermally induced micro- and macrocracks in Lac du Bonnet granite, in order to elucidate the mechanisms responsible for the temperature-dependent mechanical properties of granites. Transient heat conduction in particle-based DEM is simulated based on a network of heat reservoirs associated with each particle, and thermal pipes associated with contacts. Heat flow occurs only via conduction in the pipes that connect the reservoirs (Fig. 7.8, Wanne and Young, 2008). Each pipe is regarded as a 1D object with a thermal resistance (η) per unit length, and thus the power (Q) in a pipe is given by Q52

Ti 2 Tj ηL

(7.1)

Application of discrete element approach in fractured rock masses

Parallel bond

Heat reservoir

155

Thermal pipe

Figure 7.8 Illustration of parallel bonds, heat reservoirs, and thermal pipes in clustered particles (after Zhao, Z, Jing, L, Neretnieks, I, Moreno, L., 2014a. A new numerical method of considering local longitudinal dispersion in single fractures. Int. J. Numerical Analyt. Methods Geomech. 38, 2036 and Zhao, Z, Li, B, Jiang, Y., 2014b. Effects of fracture surface roughness on macroscopic fluid flow and solute transport in fracture networks. Rock Mech. Rock Eng. 47, 22792286). The particles with different colors represent different minerals, in order to reflect the nonhomogeneity of granites.

where Ti and Tj are the temperatures of the two reservoirs on each end of the pipe; L is the pipe length. The energy conservation equation for a single reservoir of n pipes can be expressed as 2

n X

Q 1 Qv 5 mC

1

@T @t

(7.2)

where Qv is the heat-source intensity; m is the thermal mass; and C is the specific heat. In practice, the thermal conductivity (K) is the common parameter that can be measured, and for an isotropic material, the relationship between K and η is 1 η5 2K

! 12φ X P lp Nb V b Np

(7.3)

where φ is the porosity, Vb is the disk volume, lp is the length of a thermal pipe, and Nb and Np are the numbers of disks and pipes in the volume of interest. The mechanical effect can be assumed to occur instantaneously when compared with thermal conduction. Thermal strain induced by temperature changes is considered by changing the disk radii and modifying the bond forces. Given a temperature change of ΔT, disk radius (R) is modified according to ΔR 5 αt RΔT

(7.4)

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where ΔR is the change of the radius, and αt is the coefficient of linear thermal expansion associated with the disk. Thermal strain also changes the normal force (F n ) carried by the bond, which is considered by effectively changing the bond length (L): ΔF n 5 2kn Aðαt LΔT Þ

(7.5)

where kn is the parallel bond normal stiffness, and A is the area of the parallel bond cross section. Note that L is assumed to be equal to the distance between the centroids of the two particles at the ends of the pipe associated with the bond. Lac du Bonnet granite is uniform in texture and composition, consisting, on average, of 31% quartz, 27% potassium feldspar, 38% plagioclase, and 4% mica (Jansen et al., 1993), and its thermalmechanical behavior has been investigated by laboratory tests (Lau and Chandler, 2004) and extensively modeled using the particle-based DEM (Hazzard and Young, 2004; Potyondy and Cundall, 2004; Wanne and Young, 2008). A rectangular rock sample with a height of 63.4 mm and a width of 31.7 mm was created, where four mineral groups were included with a similar mineral composition to the real granite sample (Fig. 7.9A). The average particle radius was approximately 0.23 mm. The adopted micromechanical parameters were from Potyondy and Cundall (2004). To accurately reflect the thermal behavior of rock specimens, different thermal expansion coefficients were assigned to different minerals: quartz, 24.3 3 1026 K21; potassium feldspar, 8.7 3 1026 K21; plagioclase, 14.1 3 1026 K21; and mica, 3.0 3 1026 K21 (Fei, 1995). A value of 1015 J/kg/K was taken for the specific heat for all disks, and the thermal resistance of each active thermal pipe was calculated based on a macroscopic thermal conductivity of 3.5 W/m/K and Eq. (7.3) (Wanne and Young, 2008). Two sets of scenarios were considered: continuous heating and a heatingcooling cycle (or called thermal treatment). For the continuous heating scenario, rock samples were increasingly heated from room temperature (20 C) to a peak temperature, for example, 100 C, 200 C, 300 C, and 400 C, followed by uniaxial compression or direct tension tests. For the heatingcooling scenario, the mechanical tests were performed after the heated specimens had cooled from the peak temperature to room temperature. Temperatures higher than 500 C may cause complex mineral decomposition and other chemical reactions, for example, αβ quartz phase transition, which cannot be properly modeled by the present particle mechanics model. As a result, the peak temperature was limited to 400 C, which is consistent with that for deep underground projects such as enhanced geothermal systems. To avoid provoking thermal shocking, the temperature of granite samples was assumed to change uniformly and in a sufficiently short time such that an effect can be ignored. The heating or cooling of the model was simulated by the following three-step procedure. (1) Cycle the model until a static equilibrium state, which is determined by the equilibrium ratio limit, fr (set to 1 3 1024 in this study), expressed as the ratio of maximum unbalanced for magnitude of overall particle divided by average applied force magnitude of overall particles. (2) The temperature of the granite specimens changed uniformly by 10 C. (3) Return to step 1.

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Figure 7.9 Cycling cracks in the particle-based DEM models for Lac du Bonnet granite. (A) Particle-based DEM model (Zhao, 2016) for Lac du Bonnet granite (Cho et al., 2007) and (B) thermally induced microcracks in the rock specimens after heating (upper row, black and red short line segments represent the tensile and shear cracks, respectively) and distribution of intragranular and intergranular microcracks (lower row). After Zhao, Z., 2016. Thermal influence on mechanical properties of granite: a microcracking perspective. Rock Mech. Rock Eng. 49, 747762.

Overall, both continuous heating and heatingcooling cycles can have a negative influence on the mechanical properties of granite, but the reduction in mechanical properties during heatingcooling cycles appears to be relatively small compared with continuous heating (Fig. 7.10A). This, however, cannot be understood by only considering microcrack development. Fig. 7.9B shows that microcracks started to sparsely appear after the temperature of the modeled rock specimen reached 200 C, with the number of microcracks continuously increasing with temperature up to 400 C.

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160 Heating–cooling cycle

H eating

140 120

100 80 60

Temp. (˚C) 20 100 200 300 400

40 20

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Axial stress (MPa)

120

0 0.00

0.05

0.10

0.15

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0.30

0.35

100 80 60

Temp. (˚C) 20 100 200 300 400

40 20 0 0.00

0.05

0.10

Axial strain (%) (C)

(B) 180

0.20

Max = 5 × 104 N

140 120

0.25

0.30

0.35

Max = 8 × 103 N

Diff. minerals Quartz Feldspar Plagioclase Mica

160 Number of microcracks

0.15

Axial strain (%)

Compression Tension

100 80 60 40 20 0 100–20

200-20

300-20

400–20

At 400 °C

400–20 °C

Thermal treatment

Figure 7.10 Thermal effect on the mechanical behavior of Lac du Bonnet granite. (A) Stressstrain curves during uniaxial compressive tests for numerical specimens, (B) microcrack increments during cooling phase, and (C) bond force distributions where bond forces are scaled to 5 3 104 N. After Zhao, Z., 2016. Thermal influence on mechanical properties of granite: a microcracking perspective. Rock Mech. Rock Eng. 49, 747762.

More microcracks were generated in specimens experiencing heatingcooling cycles (Fig. 7.10B), and microcracking is an irreversible process. The results show that the rock specimens with more thermally induced microcracks after heatingcooling cycles exhibited higher compressive and tensile strengths than that with fewer thermally induced microcracks from continuous heating. In Fig. 7.10C, comparison of the bond force distributions in rock specimens after continuous heating and heatingcooling cycles indicates that the thermal forces (or stresses) induced by heating decreased to low values after cooling. Therefore, significant thermal stress is likely the main reason for the drastic decrease in compressive and tensile strengths during heating, whereas thermally induced microcracking is a secondary contributor to overall decrease in strengths. This is consistent with the results of Wang et al. (2013), who also found that thermal cracking had no significant influence on brittle strength. The decrease in compressive and tensile strengths after heatingcooling cycles can, however, be attributed to the increase in the density of thermally induced microcracks.

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Figure 7.11 Thermal-gradient-induced cracks in a particle-based DEM model (after Zhao, Z., 2016. Thermal influence on mechanical properties of granite: a micro-cracking perspective. Rock Mech. Rock Eng. 49, 747762). Its side length is 150 mm and a central borehole with diameter of 10 mm is located in the center. The red and blue boundaries represent high and low temperatures, respectively. (A) Particle-based DEM model, (B) heating till 400 C, and (C) cooling from 300 to 20 C.

Heating- or cooling-induced thermal gradient can also result in rock fracturing. The disks (shown in red) close to the inner boundary of the specimen borehole were used to model heater or cooler, and hence, their temperatures were gradually changed during the modeling (Fig. 7.11A). The temperatures of the disks at the outer edges were kept constant. The results show that the macrocracks caused by a thermal gradient generally develop from cool regions toward warm regions (Fig. 7.11B and C). Shear microcracks were dominant and spatially scattered in the specimens during the early stages of heating or cooling. A threshold of the thermal gradient exists, above which macrocracks can initiate and then propagate. This is in agreement with the experiments of Jansen et al. (1993). They showed that acoustic emission events were recorded randomly throughout the rock mass during thermal cycles before the occurrence of macroscopic failure, and in the final, thermal cycle distinct clusters of acoustic emission events were observed, which were associated with radial fracturing through the rock samples.

7.4

Discrete element method for fracture networks

7.4.1 Discrete fracture network model Min et al. (2004), Baghbanan and Jing (2007), and Bidgoli et al. (2013) investigated the scale effect on the existence of elastic compliance tensor, permeability tensor, and mechanical strength in fractured rock masses at the Sellafield site, Cumbria, and England, using DEM. Their results showed that an acceptable REV scale is above 5 m for the fracture systems with constant apertures, whereas the REV size increases with the increasing second moment of the lognormal distribution of aperture for the fracture systems with correlated or uncorrelated fracture length and

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Figure 7.12 DEM model of fracture network with constant aperture and a side length of 5 m. (A) Before regulation and (B) after regulation. After Zhao, Z., Jing, L., Neretnieks, I., Moreno, L., 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Comput. Geotech. 38, 113126.

aperture. Therefore, a square DEM model (Fig. 7.12) with a side length of 5 m was extracted from the center of the original parent model of fracture system with a side length of 20 m (Zhao et al., 2013), based on field mapping results of a site characterization at the Sellafield area, undertaken by the United Kingdom Nirex Limited. In order to simplify the computations without losing generality, a few general assumptions on rock matrix, fractures, and fluid were adopted: (1) A plane-strain assumption (without considering the gravity) for stress analysis and fully saturated steady-state for fluid flow analysis; (2) The rock matrix was assumed to be impermeable, and the fluid was conducted by connected fractures only; (3) Individual fractures were idealized as the smooth parallel plate model; (4) The hydraulic apertures of fractures were equal to their mechanical apertures with constant initial values; and (5) Fluid was incompressible. Note that the following modeling results have no relation to the site geology and site investigation results or conclusions whatsoever.

7.4.2 Hydromechanical calculation For the coupled hydromechanical simulations, a 2D DEM code, called universal distinct element code (UDEC), was used to obtain the stress and deformation results of the fracture system (i.e., the fluid flow pathway changes), the changes of aperture, and flow velocity in all individual fractures contributing to the overall flow pattern. Rock blocks, treated as a linear, isotropic, homogeneous, and elastic material with constant porosity, were internally discretized into triangle finite-difference elements, whereas fracture displacements are determined by relative motions of the contacting blocks (that define the fracture concerned) in both normal and tangential directions of the fracture. In the normal direction of a parallel plate model, the constitutive model (stressdisplacement relation) is assumed to be governed by a

Application of discrete element approach in fractured rock masses

(B) Shear stress

(A)

161

σn (+)

σn3

σn2

Dilation

Unloading/reloading

σn1

(+)

ks 1 Shear displacement

un1

un2 un3

(–)

un

φd ucs Shear displacement

Figure 7.13 Mechanical behavior model of the rock fracture. (A) Normal direction and (B) shear direction. After Zhao, Z., Jing, L., Neretnieks, I., Moreno, L., 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Comput. Geotech. 38, 113126.

nonlinear stiffness, kn, in form of a simplified BartonBandis model as shown in Fig. 7.13A, Δσn 5 2kn Δun

(7.6)

In the tangential (shear) direction of the parallel plate model, the response is characterized by a constant shear stiffness, ks, before the occurrence of shear dilation (Fig. 7.13B). The shear stress, τ, is limited by a combination of cohesion and friction angle, according to a MohrCoulomb criterion. In addition, in the normal direction, the shear dilation occurs at the onset of slip (nonelastic sliding) of the fracture. Dilation is governed by the Coulomb slip model through a specified dilation angle. The accumulated dilation is generally limited by either a high normal stress level or by a critical shear displacement, ucs . Dilation is a function of the direction of shearing. Dilation increases if the shear displacement increment is in the same direction as the total shear displacement and decreases if the shear increment is in the opposite direction. To obtain the flow field in fracture networks, the mass continuity equation is solved at fracture intersections through an iterative scheme in combination with the prescribed hydraulic boundary conditions. In UDEC, a concept of “domain,” which is the region of space between blocks defined by the contact points, is used to compute the fluid pressure distribution (Fig. 7.14). The fluid flow is then governed by the pressure gradient between the adjacent domains. The flow rate per unit width of fractures follows the cubic law. The current hydraulic aperture is computed by: 2b 5 bres 1 ðunmax 2 un Þ 1 ud

(7.7)

where un and ud are normal closure (or open) and shear-induced dilation, respectively; unmax is the maximum normal displacement, approached asymptotically with

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the increasing normal stress. A residual value bres and a maximum value bmax were assumed for apertures to improve the efficiency of calculation, below or beyond which mechanical displacement does not affect the hydraulic conductivity of fractures.

7.4.3 Particle tracking method for solute transport Particle tracking method was used for transport simulations. The particles mentioned in “particle tracking” can be thought of as small solute molecules that can move by diffusion, and they are sufficiently small (about 0.3 nm) not to be strained in even the narrowest fractures. Based on an earlier computer code called “CHAN3D” developed by Moreno and Neretnieks (1993) and Gylling et al. (1999) for tracking particle movement through regular channel networks, a new code called PTFR for 2D random irregular fracture network geometries was developed and integrated with the code UDEC. Adopting the same concepts of “contact” and “domain” as that used in the UDEC data structure, the mechanical and fluid flow results obtained by UDEC calculation can be input into PTFR directly. For performing the particle tracking for transport simulation, the solute particles can be injected either along the inlet boundary or at the center of the inlet boundary. Three transport mechanisms, including advection, hydrodynamic dispersion, and matrix diffusion, can be simulated using PTFR. It is assumed that complete mixing occurred at the fracture intersections for solute transport simulations (Fig. 7.14).

Figure 7.14 Schematic diagram for “domain” and particle distribution at fracture intersection. After Zhao, Z., Jing, L., Neretnieks, I., Moreno, L., 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Comput. Geotech. 38, 113126.

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163

7.4.3.1 Advection In each fracture segment (“contact”), the water residence time, twi , is calculated by, twi 5

Vi Qi

(7.8)

where Vi is the volume of fracture segment i, that is, the product of the fracture length Li , the width Wi and the aperture bi ; Qi is the flow rate calculated using the cubic law. For 2D cases, Wi was assumed to be a unity value of 1 m. The total water residence time, t jw , for a particle (or the flow path it passed) j, is the sum of the water residence time of this particle in all the fracture segments it has passed. t jw 5

X i

twi 5

X Vi i

Qi

(7.9)

7.4.3.2 Hydrodynamic dispersion Nauman (1981) showed the probability of a particle injected at the inlet first arriving at the outlet during the time t and t 1 dt is    x 2 vt2 x f ðtÞdt 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2 4Df t 4πDf t3

(7.10)

where f ðtÞ is the probability density function of residence time of a particle through a single fracture. The cumulative distribution function of residence time is then derived as " ! !#   1 x 2 vt xv x 1 vt Erfc pffiffiffiffiffiffiffi 1 exp F ðtÞ 5 f ðtÞdt 5 Erfc pffiffiffiffiffiffiffi 2 Df 2 Df t 2 Df t 2N ðt

(7.11)

which also means the ratio of output of solute flux over the input solute flux. By the injection method, a random number uniformly distributed in the interval of [0, 1] is used to replace F(t) in Eq. (7.11), and thus the actual stochastic particle travel time through the single fracture in particle tracking simulation can be obtained by solving Eq. (7.12). ½R10

" ! !#   1 x 2 vt xv x 1 vt Erfc pffiffiffiffiffiffiffi 1 exp 5 Erfc pffiffiffiffiffiffiffi 2 Df 2 Df t 2 Df t

(7.12)

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7.4.3.3 Matrix diffusion Considering advection and matrix diffusion, the residence time distribution of a particle (flow path) can be determined by Eq. (7.13), which shows the effluent concentration, cj , for flow path j, where the water residence time is twj , under a step injection of solute with inlet concentration, c0 , at time zero (Retrock project, 2004). 0 1  j cj 1 A MPG q A 5 Erfc@ 2 Q tj 2tj 1=2 c0



t j $ t jw



(7.13)

w

where ErfcðÞ is the complementary error function; Aq is the contact surface between the flowing water and the rock, which is called the flow wetted surface (FWS). The entity of ðAq =QÞj , which is the ratio of FWS over the flow rate in the flow path, determines the interaction between the rock and the particle. The value of ðAq =QÞj is determined by 

Aq Q

j

5

X Aiq i

Qi

5

X 2Li Wi i

Qi

(7.14)

In Eq. (7.14), the ratio of ðAq =QÞj is the sum of Aiq =Qi in each fracture segment i passed by the particle, where Aiq is the FWS in fracture segment i. The value of “2” in Eq. (7.13) represents the effect of two surfaces of each fracture segment, and Wi is assumed to be a unity value of 1 m. Even though Eq. (7.13) was originally derived for a single fracture, Moreno et al. (2006) demonstrated that Eq. (7.13) is valid for a flow path consisting of a multitude of fractures. The term MPG (material property group) is assumed to be a collective-constant-containing information about effective diffusion coefficient in the matrix pores De , sorption coefficient Kd , and matrix porosity θ (Retrock project, 2004). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð1 2 θÞKd MPG 5 θ De 1 1 θ

(7.15)

Based on Eq. (7.13) that shows the cumulative probability density distribution of residence time for a flow path, the rejection method can also be used to incorporate the effects of matrix diffusion into the particle tracking. By choosing random numbers, R, from the uniform interval [0, 1] to equal to the values of cjf =c0 in Eq. (7.13), the stochastic total residence time distribution for this flow path can be determined by the following equations: 0

1  j 1 A MPG q A ½R10 5 Erfc@ 2 Q tj 2tj 1=2 w

(7.16a)

Application of discrete element approach in fractured rock masses

1 

t j 5 t jw 1  2 4 Erfc21 ½R10

  !2 MPG Aq j 2 Q

165

(7.16b)

7.4.4 Stress effects on fluid flow and solute transport in fractured rocks The numerical experiments for coupled stressflowtransport simulation were performed in the following procedure: (1) Boundary stresses were applied on the four boundaries to generate deformed models. A stress ratio is defined as K 5 horizontal:vertical stresses, and K 5 0 represents the free state that both horizontal and vertical stresses are zero. (2) Groundwater flow through the deformed DEM models was simulated under the specified hydraulic boundary condition (Fig. 7.15). The hydraulic gradient was fixed at 1 3 104 Pa/m, which was equivalent to a water head gradient of 1.0 m/m. Two sets of hydraulic conditions were simulated in order to evaluate different flow and transport behavior in macroscopically horizontal and vertical directions, respectively. (3) The fracture system geometry and fluid flow information for all fractures were transferred to PTFR, and a large number of particles were injected along or at the middle of the main inflow boundaries, that is, the right boundary for the horizontal flow case and the top boundary for the vertical flow case, respectively. Particles were collected from the other three outlet boundaries. The material properties are shown in Table 7.2.

7.4.4.1 Flow results Fig. 7.15 shows the flow pattern changes with increasing stress ratio. The flow results are presented here for understanding the solute transport results in the following subsections. In general, the hydrostatic stress condition (with equal boundary stress values in both the horizontal and vertical directions) made all the flow rates in fracture networks decrease, whereas under the larger stress difference between two discretions, a few major channels consisting of fractures undergoing significant shear dilation dominated the flow pattern, with the main flow direction parallel with direction of larger boundary stresses. If the main flow direction is perpendicular to the direction of the larger boundary stresses, the channeling followed the direction of larger boundary stresses, instead of main flow direction.

7.4.4.2 Advection Fig. 7.16 shows the breakthrough curves for conservative tracers under different stress boundary conditions. Note that the breakthrough curves account for all the particles that exited through the left, top, and bottom boundaries. Under a hydrostatic stress condition (K 5 5:5), the breakthrough curve shifted to the right direction, indicating a longer travel time of one order of magnitude from the stress-free state (K 5 0). The reason is the compression-induced decrease of flow rates in most

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Figure 7.15 Stress effects on the macroscopic flow behavior in fracture networks. (A) Two sets of hydraulic conditions were used to evaluate the anisotropic flow and transport behavior in fracture networks affected by stress. The water head gradient was fixed at 1.0 m/m. (B) Initial condition without stresses were applied. The obvious flow channels followed the main direction of water head gradient. (C) A hydrostatic stress condition (with equal boundary stress of 5 MPa in both the horizontal and vertical directions) made all the flow rates in fracture networks decrease significantly. (D) Under a larger stress difference between two discretions (horizontal and vertical stresses were 20 MPa and 5 MPa, respectively), a few major channels dominated the flow pattern, in parallel with direction of larger boundary stress. After Zhao, Z., Jing, L., Neretnieks, I., Moreno, L., 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Comput. Geotech. 38, 113126.

Application of discrete element approach in fractured rock masses

Table 7.2

Material properties of DFN models (Zhao et al., 2011) Material parameters

Intact rock

Fractures

167

Elastic modulus, E (GPa) Poisson’s ratio, υ Material property group, MPG (ms20.5) un1 (μm) σn1 (MPa) un2 (μm) σn2 (MPa) un3 (μm) σn3 (MPa) Shear stiffness, ks (GPa/m) Friction angle, ϕ ( ) Dilation angle, ϕd ( ) Cohesion, C (MPa) Critical shear displacement for dilation, ucs (mm) Initial aperture b0 (μm) Minimum aperture value, bres (μm) Maximum aperture value, bmax (μm)

Value 84.6 0.24 1 3 1028 15 4 20 10 25 30 434 24.9 5 0 3 30 5 50

of the fractures. The average particle travel length did not change significantly, but the average water residence time increased by about a factor of 5 with respect to that with K 5 0. When the stress difference between two discretions increased to K 5 15:5, the breakthrough curves started to move backwards. The main reason is that many particles chose the paths exiting from top and bottom boundaries, which resulted in the average travel length decreasing drastically. The breakthrough curve for stress ratio K 5 20:5 continued to move backwards, because of the three obvious horizontal flow channels formed under this stress condition. About 70% particles exited from the left boundary, with the normal average travel length. These fractures having larger shear dilations would dominate the flow pattern and provide the preferable particle travel paths. Fig. 7.15C shows the spatial distribution of the particle collection and flow rates normalized with respect to the maximal flow rate in the outlet fractures along the left boundary, as an example to investigate the channeling phenomenon with increasing stress ratio. Basically, fractures with larger flow rates attracted more particles. Before K 5 5:5, at least 70% particles followed the main flow direction and exited from the left boundary. When K 5 20:5, only a few fractures carried a dominated number of particles.

7.4.4.3 Hydrodynamic dispersion Zhao et al. (2014a,b) investigated macrodispersion in different fracture systems with relatively high fracture density and found that the size of the particle swarm gradually expanded and the shape of the particle swarm eventually became an

(B) 1.0

0.8

Particles input

c/c0

Particles input

c/c

(A) 1.0

0.8

0.6

0.6 K=0 K = 5:5 K = 10:5 K = 5:10 K = 15:5 K = 5:15 K = 20:5 K = 5:20

0.4

0.2

K=0 K = 5:5 K = 10:5 K = 5:10 K = 15:5 K = 5:15 K = 20:5 K = 5:20

0.4

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0.6 10

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20

20 Particle percentage

25

1.2 Particle percentage Normalized flow rate

Particle percentage Normalized flow rate

5

0.0

0 –2

–1

0

1

2

Left boundary

Figure 7.16 Stress effects on advection in fracture networks. (A) Macroscopically horizontal flow, (B) macroscopically vertical flow, (C) Particle and normalized flow rate distribution in fractures intersection the output boundary (the left vertical boundary) in macroscopically horizontal flow case (left: K 5 0; middle: K 5 5:5; and right: K 5 20:5). After Zhao, Z., Jing, L., Neretnieks, I., Moreno, L., 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Comput. Geotech. 38, 113126.

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(A) 5

5

4

4 3

2

2

1

1

Y (m)

Y (m)

3

Swarm center (–2.26, –0.50)

0

0

–1

–1

–2

–2

–3

–3

–4

–4

–5

Swarm center (–1.12, –1.28)

–5 –5 –4 –3 –2 –1 0

1

2

3

4

5

–5 –4 –3 –2 –1 0

X (m)

(B)

1

2

3

4

5

X (m)

(C) 0.6 0.5 Frequency

2

Dxx (m /s)

10–4

–5

10

K = 0:0 K = 5:5 K = 10:5 K = 15:5 K = 20:5

K = 5:5 K = 10:5 K = 15:5 K = 20:5

0.4 0.3 0.2 0.1 0.0

10–6 0

1

2

3

4

Travel distance (s)

5

6

1.0x10–5

2.0x10–5

3.0x10–5

4.0x10–5

5.0x10–5

Aperture (m)

Figure 7.17 Stress effects on hydrodynamic dispersion in fracture networks. (A) Spreading of reference particles during migration through the fracture network at t 5 60,000 s (left: K 5 5:5; right: K 5 20:5). (B) The hydrodynamic dispersion coefficients under different stress boundary conditions. (C) Fracture aperture distributions under different stress boundary conditions.

approximate ellipse during solute migration process. This behavior demonstrates that particle movement in the fracture network model can be described by a Gaussian probability distribution. Fig. 7.17 shows a comparison of shapes and sizes of the particle swarms with the varying stress ratios at t 5 1000 min. Under the condition of isotropic stress (K 5 5:5), the swarm of reference particles, moved much more slowly than the case of K 5 0, but showing a Gaussian behavior, that is, retaining the general shape of approximate ellipses during migration. The reason was the almost uniformly decreasing fracture apertures and flow rates. From K 5 0 to 5:5, the average flow rate decreased by about five times, and thus the magnitude of horizontal hydrodynamic dispersion coefficient also decreased by about five times as well (Fig. 7.17B). It means that the similar value of longitudinal dispersivity, about of 1.2 m before and after stress applied, was only dependent on fracture network geometry. When K 5 20:5, the percentage of fractures with aperture larger than the initial aperture of 30 μm increased (Fig. 7.17C). The shape of reference particle swarm became irregular (Fig. 7.17A), that is, a large number of particles followed a few

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big channels, mainly due to the shear-induced channeling. The values of horizontal hydrodynamic dispersion coefficient increased drastically, and then decreased. The reason for the decreasing equivalent hydrodynamic dispersion was that a part of reference particles following the big channels exited the DFN models.

7.4.4.4 Matrix diffusion Only the nonsorbing species (Kd 5 0) were considered in this section. With a matrix porosity of θ 5 0.316% and an effective pore diffusion coefficient of 10211 m2/s, the MPG has an approximate constant value of 1028 ms20.5. Solute particle sorption on the inner surfaces of the rock matrix can be easily added by changing the MPG, defined by Eq. (7.15). Zhao et al. (2011) showed that the effect of matrix diffusion is dependent on the flow rate, and that 80% solute particles were not influenced by matrix diffusion under a hydraulic gradient of 104 Pa/m (1 m/m). Note that in reality the hydraulic gradient at a site for radioactive waste repositories is most likely several orders of magnitude smaller. According to the cubic law, the flow rate in a fracture is just scaled by the hydraulic gradient. In this way, the flow rate can be simply decreased by a factor of 1000 in each fracture to obtain new flow field under a much decreased hydraulic gradient of 10 Pa/m, based on the previous case of 104 Pa/m. Then, the solute transport simulations considering matrix diffusion were performed for the lower hydraulic gradient condition. The trends of the breakthrough curves shifting with increasing stress ratio were similar between interacting (Fig. 7.18) and conservative tracers (Fig. 7.16). However, the shapes of breakthrough curves became less steep and with much longer tails. This is because (1) solute particles can randomly enter the stagnant water in the rock matrix by molecular diffusion and reside there for some time before they again move out to the flowing water in the same or other fractures buy diffusion, and thus solute particles always arrive later than the water residence time and have a distribution of residence time rather than a constant water residence time; (2) a small fraction of

Figure 7.18 Stress effects on advection in fracture networks. (A) Macroscopically horizontal flow and (B) macroscopically vertical flow. After Zhao, Z., Jing, L., Neretnieks, I., Moreno, L., 2011. Numerical modeling of stress effects on solute transport in fractured rocks. Comput. Geotech. 38, 113126.

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particles, meandering through more tortuous flow paths with extremely large, resided within the matrix for a much longer time.

7.4.5 Effects of roughness on macroscopic flow and transport in fractured rocks Rock fractures are typically rough-walled, and fracture surface roughness has significant impact on fluid flow and transport processes in single fractures. In other words, the mechanical aperture of a rock fracture is usually larger than its hydraulic aperture. However, the effects of this reduced hydraulic aperture (due to roughness) in local fractures on macroscopic fluid flow and solute transport in complex fracture systems are still not clear, and most previous studies assumed the identical hydraulic and mechanical apertures in DFN models for simplicity (Min et al., 2004; Zhao et al., 2011). In this section, the influences of local surface roughness of fractures on fluid flow and solute transport processes at the macroscopic scales of fracture networks is preliminarily investigated. Li and Jiang (2013) gave a relationship between mechanical aperture (E) and hydraulic aperture (e), e5 e5

E 1 1 Z22:25 1 1 Z22:25

ðRe , 1Þ E  1 0:00006 1 0:004Z22:25 ðRe 2 1Þ 

(7.17a)

ðRe $ 1Þ

(7.17b)

where Z2 is a dimensionless parameter to characterize the fracture surface roughness; Re is the Reynolds number that is the ratio of inertial forces to viscous forces and quantifies the relative importance of these two types of forces for given flow conditions. Note that, Reynolds number is usually small for groundwater system, so only Eq. (7.16a) was used in the following calculations. Fracture roughness may vary within different fractures. Three artificial cases of roughness distribution, including constant Z2, uniform distribution of Z2, and normal distribution of Z2, were considered, in order to obtain a generic understanding. The minimum and maximum of Z2 was assumed to be in the range (0.1, 0.5), which roughly represents Joint Roughness Coefficient (JRC) changing from 1 to 20 (Li and Jiang, 2013). The case without considering roughness (“parallel plate” model) was also considered as reference. Considering a macroscopically horizontal flow through the fracture network, obvious channels with different magnitudes of flow rates were observed in all the cases (Fig. 7.19A). The model with uniform fracture roughness (Z2 5 0.5) exhibited the smallest total flow rate, and the total flow rate decreased by 47%, compared with the model without considering roughness (Z2 5 0). The flow rates in the other two models of uniform and normal roughness distributions were similar, which were bounded by the fracture networks with constant zero roughness (Z2 5 0) and largest roughness (Z2 5 0.5).

Figure 7.19 Effects of fracture surface roughness on macroscopic fluid flow and solute transport in fracture networks. (A) Flow rate distributions fracture networks with varying roughness distributions and (B) comparison of breakthrough curves for advection (left) and advection plus matrix diffusion (right). After Zhao et al. (2014).

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Based on the flow fields, solute transport was modeled using particle tracking method. The shapes of breakthrough curves in the different cases were almost the same (Fig. 7.19B), because of the similar flow patterns but different magnitudes in all the models. The fracture roughness just changed the positions of breakthrough curves (Fig. 7.5). The average travel distances were almost the same for models with constant roughness, but the models with uniform or normal distributions of roughness exhibited slightly longer mean travel distances. With increasing roughness (Z2), the breakthrough curves shifted to the longer time direction, and the average water residence time increased. For matrix diffusion simulation, we only considered nonsorbing species (Kd 5 0, MPG 5 1028 m/s0.5). The ratio of FWS over flow rate (Aq/Q) determines the interaction between rock matrix and solute particles. A larger value of Aq/Q allows the solute particles to access the saturated stagnant water in the matrix pores and, thus, gives the particle a longer time to reside in the matrix. With increasing roughness, the average values of Aq/Q increased significantly, and the longer residence time was predicted. Generally, the breakthrough curves for the models with uniform and normal distributions of roughness were close to each other. By considering different roughness distributions, the results showed that fracture roughness can affect the flow rates and residence time to some extent. However, fracture roughness has negligible influences on the general patterns of fluid flow (channeling) and solute migration (e.g., travel distances and shape of breakthrough curves). This is because the connectivity of a fracture network is determined by other geometrical parameters (fracture positions, orientations, and lengths), rather than roughness or apertures. Whether fracture roughness should be considered for a specific site depends on the magnitudes and distribution of roughness parameters.

7.5

Summary remarks

Using DEMs to model the coupled THMC processes in single fractures and fractured rock masses are presented, respectively. The main conclusions are summarized below: 1. Particle-based DEM is a robust tool to model the mechanical behavior of rock matrix and single fractures. Fracture surface damage occurs along fracture skin layers during shear, and the newly generated gouge particles and microcracks in the damaged layers are strongly dependent on normal stresses and shear displacements. Gouge particles can roll with the moving fracture walls or be can be crushed into a few major pieces and a large number of minor comminuted particles, which is also strongly dependent on normal stresses and shear displacements. 2. Particle-based DEM is capable of modeling the thermal cycling cracks and thermal gradient-induced cracks in rock matrix. Heating generally reduces the compressive and tensile strengths of granites, first because of increasing thermal stresses, and second because of the generation of tensile microcracks. The presence of a thermal gradient induces the formation of macrocracks, which propagate from relatively cool to relatively warm areas. 3. A code called PTFR was developed to simulate the solute transport in 2D irregular fracture systems using the particle tracking method, based on the coupled stress-flow results from UDEC. The main effect of stress is through its influence on fracture aperture change

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that changes fluid flow velocity in turn. This change will then affect the travel path and residence time of solute particles. 4. Fracture roughness can affect the flow rates and residence time to some extent, but it has negligible influences on the general patterns of fluid flow and solute migration. For a first-order approximation, local fracture roughness may be neglected, but for an accurate estimation, local fracture roughness should be considered.

Acknowledgments This work was financially supported by the National Basic Research Program of China (973 Program, 2014CB047003), the National Natural Science Foundation of China (No. 51509138), Beijing Natural Science Foundation (No. 8152020), the Recruitment Program of Beijing Youth Experts (2014000020124G115), the Open-end Research Fund of the State Key Laboratory of Water Resources and Hydropower Engineering Science (No. 2014SGG04) and Young Elite Scientist Sponsorship by China Association for Science and Technology. Part of the work in this chapter was done when the author studied at Royal Institute of Technology (KTH), Sweden, and worked at Stockholm University. SKB, the Swedish Nuclear Fuel and Waste Management Company, is gratefully acknowledged for financial support through DECOVALEX-2011 project. Financial support from the Bolin Center for Climate Research at Stockholm University is also acknowledged. Dr. Lanru Jing, Prof. Alasdair Skelton, Prof. Ivars Neretnieks, Prof Luis Moreno, Prof Er-xiang Song, Prof Yifeng Chen, Prof Bo Li and Prof Dawei Hu are thanked for constructive comments and discussion.

References Asadi, M.S., Rasouli, V., Barla, G., 2012. A bonded particle model simulation of shear strength and asperity degradation for rough rock fractures. Rock Mech. Rock Eng. 45, 649675. Baghbanan, A., Jing, L., 2007. Hydraulic properties of fractured rock masses with correlated fracture length and aperture. Int. J. Rock Mech. Min. Sci. 44, 704719. Berkowitz, B., 2002. Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour. 25, 38613884. Berkowitz, B., Bear, J., Braester, C., 1988. Continuum models for contaminant transport in fractured porous formations. Water Resour. Res. 24, 12251236. Bidgoli, M.N., Zhao, Z., Jing, L., 2013. Numerical evaluation of strength and deformability of fractured rocks. J. Rock Mech. Geotech. Eng. 5, 419430. Billaus, D., Chiles, J.P., Hestir, K., Long, J., 1989. Three-dimensional statistical modelling of a fractured rock mass—an example from the Fanay-Auge`res mine. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 26, 281299. Cho, N., Martin, C.D., Sego, D.C., 2007. A clumped particle model for rock. Int. J. Rock Mech. Min. Sci. 44, 9971010. Cundall, P.A., 1999. Numerical experiments on rough joints in shear using a bonded particle model. In: Lehner, F.K., Urai, Jl (Eds.), Aspects of tectonic faulting. Springer, Berlin, pp. 19.

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Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique. 29, 4765. de Dreuzy, J.R., Davy, P., Bour, O., 2000. Percolation parameter and percolation-threshold estimates for three-dimensional random ellipses with widely scattered distributions of eccentricity and size. Phys. Rev. E. 62, 59485952. Doughty, C., 1999. Investigation of conceptual and numerical approaches for evaluating moisture, gas, chemical, and heat transport in fractured unsaturated rock. J. Contam. Hydrol. 38, 69106. Fei, Y., 1995. Thermal expansion. In: Ahrens, T.J. (Ed.), Mineral physics and crystallography: a handbook of physical constants. AGU, pp. 2944. Gylling, B., Moreno, L., Neretnieks, I., 1999. The channel network—a tool for transport simulations in fractured media. Ground. Water. 37, 367375. Hazzard, J.F., Young, R.P., 2004. Dynamic modelling of induced seismicity. Int. J. Rock Mech. Min. Sci. 41, 13651376. Hofmann, H., Babadagli, T., Zimmermann, G., 2015. A grain based modeling study of fracture branching during compression tests in granites. J. Rock Mech. Min. Sci. 77, 152162. Itasca Consulting Group Inc., 2008. PFC2D User’s Guide. Minneapolis, MN. Ivars, D.M., Pierce, M.E., Darcel, C., Reyes-Montes, J., Potyondy, D.O., Young, R.P., et al., 2011. The synthetic rock mass approach for jointed rock mass modelling. J. Rock Mech. Min. Sci. 48, 219244. Jackson, C.P., Hoch, A.R., Todman, S., 2000. Self-consistency of a heterogeneous continuum porous medium representation of a fractured medium. Water Resour. Res. 36, 189202. Jansen, D.P., Carlson, S.R., Young, R.P., Hutchins, D.A., 1993. Ultrasonic imaging and acoustic emission monitoring of thermally induced microcracks in Lac du Bonnet Granite. J. Geophys. Res. 98, 2223122243. Jing, L., Hudson, J., 2002. Numerical methods in rock mechanics. Int. J. Rock Mech. Min. Sci. 39, 409427. Jing, L., Stephansson, O., 2007. Fundamentals of discrete element methods for rock engineering—theory and application. Elsevier Science Publishers, Rotterdam, p. 562. Jing, L., Min, K.-B., Baghbanan, A., Zhao, Z., 2013. Understanding coupled stress, flow and transport processes in fractured rocks. Geosyst. Eng. 16, 225. Lau, J.S.O., Chandler, N.A., 2004. Innovative laboratory testing. Int. J. Rock Mech. Min. Sci. 41, 14271445. Li, B., Jiang, Y., 2013. Quantitative estimation of fluid flow mechanism in rock fracture taking into account the influences of JRC and Reynolds number. J. MMIJ. 129, 479484 (In Japanese). Lisjak, A., Grasselli, G., 2014. A review of discrete modeling techniques for fracturing processes in discontinuous rock masses. J. Rock Mech. Geotech. Eng. 6, 301314. Liu, H.H., Haukwa, C.B., Ahlers, C.F., Bodvarsson, G.S., Flint, A.L., Guertal, W.B., 2003. Modeling flow and transport in unsaturated fractured rocks: an evaluation of the continuum approach. J. Contam. Hydrol. 6263, 173188. Long, J.C.S., Remer, J.S., Wilson, C.R., Witherspoon, P.A., 1982. Porous media equivalents for networks of discontinuous fractures. Water Resour. Res. 18, 645658. Mair, K., Abe, S., 2008. 3D numerical simulations of fault gouge evolution during shear: grain size reduction and strain location. Earth. Planet. Sci. Lett. 274, 7281. Min, K.-B., Jing, L., Stephansson, O., 2004. Fracture system characterization and evaluation of the equivalent permeability tensor of fractured rock masses using a stochastic REV approach. Hydrogeol. J. 12, 497510. Moreno, L., Neretnieks, I., 1993. Fluid flow and solute transport in a network of channels. J. Contam. Hydrol. 14, 163192.

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The embedded finite element method (E-FEM) for multicracking of quasi-brittle materials

8

Paul Hauseux1, Emmanuel Roubin2 and Jean-Baptiste Colliat1 1 University of Lille, Lille, France, 2University of Grenoble, Grenoble, France

8.1

Introduction

Dealing with brittle or quasi-brittle failure within a finite element (FE) context is a well-known problem and basically requires to implement new methods since the finite element method (FEM) has originally been thought for continuum media, i.e., without discontinuities. The models using traditional FEM and based on the linear fracture mechanics require mesh refinements in order to model the singularity of the stress field around the crack tip. Several numerical schemes have been used in this view: the double noding technique (Kanninen et al., 1982; Liaw et al., 1984), local or general remeshing (Rashid, 1998; Seyedi et al., 2006), boundary elements (Aliabadi, 1997), cohesive zone models (Charlotte et al., 2006), smeared cracks models (Pijaudier-Cabot and Bazant, 1987; Mazars and Pijaudier-Cabot, 1989), and models with embedded discontinuities (Moe¨s et al., 1999; Oliver, 1996a; Wells and Sluys, 2001; Jira´sek, 2000; Simo et al., 1993). The latter family is the most physically based because cracks are nothing else but real discontinuities within the material itself. Moreover the released energy corresponding to the crack opening process is dissipated onto the discontinuity. Thus no mesh dependency is observed and no numerical treatments of the strain localization are required. Their numerical implementations can be made according to two main ways: the extended finite element method (X-FEM) (Mo¨es et al., 1999), which corresponds to a global kinematic enrichment, and the embedded finite element method (E-FEM) (Ortiz et al., 1987), which is a local enhancement. Historically, smeared crack models are the first being developed. For instance, damage models belong to this family and are still very popular within the civil engineering community. Thanks to their local aspect, they are very easy to implement and allow for large-scale structural simulations. In a continuous analysis, crack initiation can be theoretically studied as a bifurcation problem (Hill, 1958; Rudnicki and Rice, 1975). For constitutive laws with softening, the problem is numerically ill-posed and a strong dependence of the FE solution to the mesh size and orientation is observed (Pietruszczak and Mro´z, 1981; Bazant and Belytschko, 1985). Here nonlocal approach is a solution to such issue (Pijaudier-Cabot and Bazant, 1987; Peerlings et al., 1996). Yet, accurate transition from damage onset to crack initiation and propagation remains a scientific challenge. Moreover, these Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00008-7 © 2017 Elsevier Ltd. All rights reserved.

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models provide little information about cracks patterns, openings, orientations, etc. Thus models with kinematic enhancements have emerged. By adding a discontinuity within the displacement field (“strong discontinuity”), they allow to represent a fracture as a real “gap.” A review by Oliver et al. (2006) stresses on the differences between X-FEM (intrusive) and E-FEM (nonintrusive) implementations. The main advantage of the X-FEM is to allow for a rich representation of the stress field near the crack tip, which is not possible with the E-FEM. This lack of phenomenological representativity is known to be the reason of problems such as stress locking (Jirsek, 2012) which leads to unrealistically high structural strength. Additional constrains such as imposing the crack path continuity help overcome this issue (Oliver et al., 2014). With these solutions, E-FEM-based implementations are known to be able to address quasi-brittle failure problems at the structural scale in the context of multiscale resolutions (Oliver et al., 2015) or dynamic loading (Lloberas-Valls et al., 2016) (not treated in this chapter). Finally, the E-FEM is especially well adapted to contexts where multicracking occurs. Indeed, by using the so-called static condensation procedure (Wilson, 1974), the E-FEM preserves the overall size of the discrete problem, whatever the number of discontinuities (and so the number of cracks) is. Hence the E-FEM is rather cheap since the memory needed is independent of the number of crack. Furthermore, from a practical point of view, the nonintrusive aspect of E-FEM leads to a simple and straightforward implementation within any FE code (provided that one can access to the “constitutive law” routine). During the last two decades, two-dimensional (2D) kinematic enhancements with pure opening were proposed and widely used (Brancherie and Ibrahimbegovic, 2009). Efforts have also been spent for 3D models and such models are nowadays available, even dealing with complex kinematics such as sliding. Thus, the goal of this chapter is to present and investigate these 3D complex kinematics.

8.2

Overview of the E-FEM

The goal of the E-FEM is to provide a non intrusive (and so quite easy) way to embed a strong discontinuity (i.e., within the interpolation of the displacement field) of a single element. The main ingredients for that are (1) the kinematics of this discontinuity (i.e., how the interpolation space is enriched from the classical FEM), (2) the new variational form of the equilibrium problem that provides additional equations corresponding to the new elementary unknowns, (3) a simple resolution procedure mimeting the operator-split procedure, which is well known for solving FE models with material nonlinearities such as plasticity (Simo et al., 1993).

8.2.1 Kinematics description of strong discontinuities The kinematics of strong discontinuities within an FEM framework has been introduced by Simo et al. (1993). Further developments can be found by Simo and

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Oliver (1994), Oliver (1996a), and Wells and Sluys (2001), where numerical implementations are made in 2D and 3D spaces. Basically, an element Ωe is considered to be split by a plane surface Γd , representing a discontinuity in the displacement field. If u is taken to be a smooth function over Ωe representing the regular part of the displacement field and ½juj a piecewise constant function representing the displacement jump, this discontinuity can be modeled by decomposing a theoretical displacement field u into a regular and an enhanced part such as follows: u ¼ u þ HΓd ½juj

(8.1)

where HΓd is the Heaviside function centered on Γd (unit valued in Ωe and null in Ωe ). Within a FE framework, such interpolation function inevitably leads to an illformulated problem, regarding the displacement boundary conditions. Following Oliver (1996b), the introduction of an arbitrary continuous function ϕe of unit value at each node in Ωe and null at each node in Ωe can overcome this issue. By establishing an arbitrary displacement function u as: u ¼ u þ ϕe ½juj

(8.2)

the theoretical displacement field Eq. (8.1) can be rewritten as: u ¼ u þ ðHΓd 2 ϕe Þ½juj

(8.3)

Eqs. (8.1) and (8.3) are strictly equivalent. Nevertheless, using the latter eventually leads to an interpolation of (HΓd 2 ϕe ), which can be performed using functions of zero nodal values. Furthermore, the displacement unknowns are u whose nodal values carry the displacement jump information (through ϕe ). An example in one dimension with ϕe taken as a linear function is depicted on the previous sketch, where Fig. 8.1A shows the construction of enhanced part interpolation function and Fig. 8.1B shows the decomposition of the discontinuous displacement u where u is the regular and continuous part of the displacement field that allows to impose standard boundary conditions (Oliver, 1996b) and HΓd is the Heaviside function cen2 tered on Γd , i.e., unit value in Ωþ e and null in Ωe . ϕe is an arbitrary function whose þ value is equal to one for each node in Ωe and zero for each node in Ω2 e (Wells and Sluys, 2001). The strain field can be written as: E ¼ rs u þ E^

(8.4)

where E^ is the part related to the strong discontinuity, i.e. E^ ¼ ðHΓd 2 ϕe Þrs ½juj 2 ð½juj  rs ϕe Þs þ δΓd ð½juj  nÞs |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} E^ b

(8.5)

E^ u

The enhanced strain field E^ can be decomposed into two parts: regular and bounded part E^ b and an unbounded one E^ u . δΓd is the Dirac-delta distribution

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Figure 8.1 1D representation of discontinuous displacement decomposition: (A) decomposition of the enhancement function and (B) decomposition of the discontinuous displacement. Source: Oliver, J. 1996b. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals. Part 2: Numerical simulation. Int. J. Num. Meth. Eng. 39 (21), 35753623.

centered at the surface discontinuity. From a physical point of view, Eq. (8.5) leads to localize strains at the discontinuity. Taking ½juj like a constant on the interface Γd of the element, the first term containing rs ½juj vanishes. This leads to an explicit formulation of the bounded part of the strong enhanced strain referred as kinematically enhanced strain (KES) (Roubin et al., 2015): E^ b 5 2 ð½juj  rs ϕe Þs

(8.6)

The term E^ u is calculated on the domain Ω\Γd using a constitutive law at the interface level (MohrCoulomb criterion for example) that provides a link between stress and displacement jump (discrete strong discontinuity approach; Oliver, 2000). The three-field variational formulation (Washizu, 1982) justifies the use of E-FEM in a classical FEM context where all fields ðu; E; σÞ, respectively the standard displacement field, the standard strain field, and the standard stress field, are considered independent. The set ðη; γ; τÞ refers to the virtual standard displacement field, the virtual standard strain field, and the virtual standard stress field, respectively. The three-field formulation system is described using the enhanced assumed strain method (EAS) developed by Simo and Rifai (1990). Firstly, standard and virtual strain fields are enhanced in the same way. η is the regular part of the standard

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displacement field. Then, it is assumed that the space of the stress field is L2 -orthogonal to the space of the enhanced strains. After imposing the L2 -orthogonal condition, the EAS must ensure that an enhanced element still satisfies the patch test after imposing the orthogonal condition (more details can be found by Roubin et al. (2015)). Hence, the stress field must at least include piecewise constant and with the combination of the L2 -orthogonal condition, the resulting equation reads: ð

ð

ð Ωe

γ^ dΩ ¼

Ωe

γ^ b dΩ þ

Ωe

γ^ u dΩ ¼ 0

(8.7)

γ^ is the part of γ related to the strong discontinuity. Note that γ^ respects the form of ε^ , thus it is decomposed into a bounded part γ^ b and an unbounded part γ^ u . If we choose for γ^ u a form respecting E^ u , we obtain: ð

ð Ωe

γ^ b dΩ þ

Γd

ð½jηj  nÞs dΩ¼ 0

(8.8)

The same assumption of constant strain field and flat interface within an element are made. Thus, γ^ b has the form of the so-called EAS: γ^ b 5 2

A ð½jηj  nÞs V

(8.9)

We use in this work KES and EAS. This solution is called statically and kinematically optimal nonsymmetric formulation (SKON formulation) (Dvorkin et al., 1990).

8.2.2 Hu-Washizu formulation Having the kinematics in hands, we now turn to the variational formulation of the equilibrium problem. In contrast to standard displacement formulations, where the equilibrium equation is expressed in its weak form, the Hu-Washizu (Washizu, 1982) three-field variational formulation provides a suitable unified mathematical statement for enhanced FE kinematics. The main idea is to consider displacement, stress, and strain fields—noted u, σ, and ε, respectively—as independent. Hence, each fundamental equations of the mechanical problem are formulated in its weak form, using three corresponding virtual fields—η, τ, and γ, respectively. The variational statement of the equilibrium equation of the stress field, the kinematics relationship between the strain and displacement field, and constitutive model— Eqs. (8.10a), (8.10b), and (8.10c), respectively—are expressed as follows: Find ðu; ε; σÞ A ðV; E; TÞ so that ’ ðη; γ; τÞ A ðV0 ; E; TÞ, ð HWu ðu; ε; σ; ηÞ ¼

ð Ω

rs η:σ dΩ 2

Ω

ð η  b dΩ 2

Γt

η  t d@Ω ¼ 0

(8.10a)

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ð HWσ ðu; ε; σ; τÞ ¼

τ:ðrs u 2 εÞ dΩ¼ 0

(8.10b)

γ:ðσðεÞ  2 σÞ dΩ¼ 0

(8.10c)

Ω

ð HWε ðu; ε; σ; γÞ 5

Ω

with V ¼ fujuAH 1 ðΩÞ; u ¼ u on Γu g; u ¼ 0 on Γu g and

V0 ¼ fujuAH 1 ðΩÞ;

E ¼ fε j εAL2 ðΩÞg

T ¼ fσjσAL2 ðΩÞg

where t is the traction vector imposed on the Neumann surface Γt , b the internal force, and σ a stress field that respects the constitutive law on Ω. It is worth noting that, dealing with usual displacement formulations, both kinematics relationship and constitutive model are supposed to be verified in a strong sense, i.e., σ ¼ σðεÞ  and ε ¼ rs u, making Eqs. (8.10b) and (8.10c) irrelevant. On the one hand, additional unknowns have been brought by kinematics enhancements and on the other hand, the presented formulation brings additional equations to the system. However, it remains many ways to solve it. Since literature is well documented on the matter, the different possibilities are simply enunciated, focusing only on the critical points. Full description of the methods can be found by Jira´sek (2000) and in the other citations that follow. Two major solutions of enhancement (based on the one hand, on statistical considerations and on the other hand, on kinematics considerations) assist in the resolution of the Hu-Washizu formulation system. Note that, at this stage, a discretization is necessary and equations have to be considered within an FE context. Several ways have been explored in the literature:

8.2.2.1 The statically optimal symmetric formulation First statically optimal symmetric (SOS) formulation can be found by Belytschko et al. (1988). Then, this work has been enhanced by a great deal of contributions over the next decades. Among them, Larsson et al. (1996), Armero and Garikipati (1996), and Sluys and Berends (1998). The main idea of this method is to consider that the interpolation of the displacement field is not enhanced. Basically, it results in a compatibility condition between the stress interpolation matrix and the enhanced strain. An important assumption of L2 -orthogonality is made between the virtual strain field and the actual stress field. Under this assumption, the latter vanishes from the formulation, eventually leading to a so-calledÐzero mean condition onto the Ð enhanced part of the strain field Ω ε~ dΩ ¼ 0 and Ω ε^ dΩ ¼ 0. Hence, the still unknown shape of the enhanced part can be determined. Based on purely statistical considerations (patch test), a construction of the interpolation matrix of strain enhancement respecting the zero mean condition brings continuity to the stress field.

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Furthermore and due to the fact that strain fields and their variations are interpolated the same way, the resulting matrix of the system is proved to be symmetrical. The major drawback of this enhancement construction is the lack of kinematical meaning. Indeed, no discontinuities (weak nor strong) are kinematically represented.

8.2.2.2 The kinematically optimal symmetric formulation The lack of kinematical representation can be dealt with by modifying the enhanced interpolation process. Although for SOS it directly derives from the three-field formulation, kinematically optimal symmetric (KOS) proposes to construct it regarding only a meaningful description of discontinuity kinematics (Lotfi and Shing, 1995). Hence, displacement field is enhanced by a suitable interpolation matrix. The corresponding strain is therefore deduced, applying the symmetrical kinematics operator rs . It occurs that the resulting system is strictly identical to the one obtained by using SOS formulation (stress field deleted from the equations and symmetric matrix system). The only differences reside in the construction of the enhanced strain part interpolation. Herein, the zero mean condition is not respected but kinematics of discontinuities is well represented.

8.2.2.3 The SKON formulation First brought out by Dvorkin et al. (1990), followed by Simo and Oliver (1994), and fully developed by Oliver (1996a) (but not called so at the time), the SKON formulation takes the advantages of both methods. Actual and virtual strain fields are not interpolated the same way. When the latter respects the zero mean condition of SOS formulation, the suitable representation of discontinuity kinematics is chosen for standard enhanced strain field (KOS), leading inevitably to a nonsymmetrical system.

8.2.3 FE discretization As seen previously, Hu-Washizu formulation and assumed strain method allow us to use E-FEM in a classical context of FEMs. Discretization of the standard strain field (Eq. (8.6)) and the virtual strain field (Eq. (8.9)) is not written in the same way. Indeed, as stated in Section 8.2.1, the virtual strain field is based on EAS (Eq. (8.9)) and the standard strain field is based on KES (Eq. (8.6)). This nonsymmetric formulation gives better numerical results (Wells and Sluys, 2001). The enhanced strain fields are discretized with Voigt notations as:  T B d þ Gs ½juj ε ¼ εxx εyy εzz 2εxy 2εyz 2εxz ¼ |{z} |fflfflffl{zfflfflffl} rs u

ε^ b

 T γ ¼ γxx γ yy γ zz 2γ xy 2γ yz 2γ xz ¼ |{z} B δ þ Gs ½jηj |fflfflffl{zfflfflffl} rs η

(8.11a)

γ^

(8.11b)

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where Gs (dimension of B) is the matrix equivalent to the operator ð  rϕe Þs , Hs (simple operator, nondimensional) equivalent to the operator ð  nÞs , and, according to Eqs. (8.9) and (8.5), the EAS interpolation matrix Gs is calculated as:   A    (8.12) Gs ¼ Gs;b þ Gs;u ¼ 2 þ δΓd Hs V with a bounded part Gs;b and an unbounded part Gs;u . Bð¼ rNÞ is the standard interpolation matrix of conventional FEMs and d the vector which contains the displacement of each node. The function ϕe can be built from the conventional shape functions of the first order: ϕe ðxÞ ¼

nelm X

N a pa

(8.13)

a¼1

with pa ¼ 1 for each node in Ωþ e and zero otherwise. By inserting Eqs. (8.11) into the three-field formulation system, the discretization of the problem leads to the following global system (Roubin et al., 2015): e e elm Ane¼1 ½fint 2 fext ¼ 0

(8.14)

and for all elements, A h½juj 5 2 V

ð Ωe

Hs;T σ



ð

dΩ þ



Γd

Td@Ω

¼0

(8.15)

with σ the stress field which verifies the constitutive law σ ¼ CðBd þ Gs ½jujÞ and T ¼ Hs;T σjΓ  d the traction vector at the interface defined from the “traction-separation” law from Γd . Eq. (8.15) reflects the continuity of the traction vector at the interface Γd . Internal and external forces are calculated conventionally: ð e fint ¼

ð Ωe

BT σ dΩ

and

e fext ¼

Ωe

ð N T b dΩ 2

Γt

N T t@Ω

(8.16)

Eq. (8.14) is the global equilibrium equation of a problem as the standard equation (Eq. (8.15)) is a local equation for each element. The strong discontinuity ð½jujÞ is an internal variable. It is determined by a local resolution within each element and it does not change the overall size of the system. The strong discontinuity ½juj can be written as a scalar ½u, which represents the value of the sliding/opening multiplied by a unit vector np , which gives the direction and the orientation of the sliding/opening: ½juj ¼ ½u np

(8.17)

The discrete model is based upon a relationship between the traction vector T and the crack opening/sliding magnitude labeled ½u. A failure criterion makes it

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possible to know when cracks are initiated. As long as this one is negative, strong discontinuities are not activated ð½juj ¼ 0Þ and the resolution of only Eq. (8.14) is classically made, i.e., resolution of a linear problem. From the time there is localization, an opening criterion Φo ¼ 0 connects the traction vector T and the crack opening/sliding. Then, we are faced to a coupled system where Eq. (8.14) is the nonlinear global system whose unknowns are contained in d and Eq. (8.15) is a nonlinear local system with additional unknowns ½u (h½juj ¼ 0 for each element). d is calculated with a static condensation (Wilson, 1974) on the local (known) variables ½u by solving the global nonlinear system. Thus, the overall size of the system is not increased. For some criteria, the value of ½u can be obtained analytically and so, it is possible to avoid a local Newton resolution to solve the local system. Generally, to solve the nonlinear system Eq. (8.14), an iterative solver is implemented like Newton or BroydenFletcherGoldfarbShanno (BFGS) methods (Matthies and Strang, 1979). A MohrCoulomb criterion is proposed to characterize shear fractures (fracturing in mode II). The next paragraph presents the integration of this law within the E-FEM context with four-node tetrahedron elements.

8.3

Application to induced fracture networks around drifts after an excavation in claystone

We present both the integration of a MohrCoulomb criterion for sliding and an application of the whole framework to the fracturing process of claystone during an excavation.

8.3.1 Model description The kinematic description of strong discontinuities described in Section 8.2 is here used to model shear fractures, i.e., mode II propagation, with an anisotropic MohrCoulomb criterion (Hauseux et al., 2016). The MohrCoulomb criterion is very commonly used for rocks. It introduces a cohesion C and a friction angle ϕ. The MohrCoulomb failure criterion Φl can be written as: Φl ¼ supðOTt O þ Tn tan ϕ 2 CðnÞÞ

(8.18)

n

where Tt and Tn ¼ TnUn are the projections of the traction vector T at the interface Γd with normal vector n such as T ¼ Tn þ Tt . When C and ϕ are constant, it is possible to analytically derive the coordinates of the normal vector of the shear plane which maximizes the criterion (Bai and Wierzbicki, 2009). In the eigenbasis, n has coordinates ðν 1 ; ν 2 ; ν 3 Þ and we denote c1 ¼ tanϕ. Then, we get the following expressions for OTt O and Tn : OTt O ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ν 21 ν 22 ðσ1 2σ2 Þ2 þ ν 22 ν 23 ðσ2 2σ3 Þ2 þ ν 21 ν 23 ðσ3 2σ1 Þ2

(8.19)

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Tn ¼ ν 21 σ1 þ ν 22 σ2 þ ν 23 σ3

(8.20)

with 8 1 > > pffiffiffiffiffiffiffiffiffiffiffiffiffi ν 21 ¼ > > > 1 þ ð 1 þ c21 2c1 Þ2 > < ν 22 ¼ 0 > > 1 > > ν2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi > 3 > : 1 þ ð 1 þ c21 2c1 Þ2 In an anisotropic case, where the cohesion depends on the orientation of the sliding plane (for anisotropic materials like most of sedimentary rocks), it is necessary to numerically solve the maximization problem (8.18). The constitutive law is written as follows: σ ¼ CE þ σ0

(8.21)

with in situ initial stress field σ0 and stiffness matrix C. Cracks are initiated when the failure criterion reaches zero. Then, the link between the components of the traction vector OTt O, Tn , and the crack opening/sliding ½u is made by the following opening criterion: Φo ¼ OTt O þ Tn tan φ 2 CðnÞ  expð2 CðnÞ½u=Gf Þ ¼ 0

(8.22)

where Gf (J  m22 ) is the fracture energy of the material. This parameter controls the rate of decrease of the traction separation vector at the discontinuity. It is worth noting that the existence of this parameter is fully related to the dissipation of energy onto the surface of discontinuity. This leads to a well-posed problem and is a major advantage of the strong discontinuity approach. After failure, the following assumption for the opening criterion is made: OTt O  TtUnt . n is the normal vector of the shear plane and nt is the unit vector in the same direction and orientation of Tt at the localization. Fig. 8.2 shows an element with an embedded strong discontinuity (mode I and mode II).

Figure 8.2 Element with embedded strong discontinuity (mode I and mode II kinematics). ½juj is a vector. Element (A), mode I (B), mode II (C).

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For shear fractures (fracturing in mode II), the strong discontinuity comes from the slid2 ing on the domain Ωþ e of Ωe (Fig. 8.2C) and ½u is the sliding magnitude. So, np is a linear combination of two orthogonal vectors m and t belonging to the interface, this vector is denoted nt . We have for Tn and OTt O: Tn

¼

nT Hs;T σ

¼

nT Hs;T CBd þ nT Hs;T σ0 þ nT Hs;T CGs np ½u |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} δ

OTt O

¼ ¼

TtT nt ¼

ξ

OHs;T σ 2 Tn nO nTt Hs;T σ0 þ Hs;T CBd  nt þ nTt Hs;T CGs np ½u |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} α0

(8.23)

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} β0

We obtain finally the following expression for Φo : Φo ¼ α0 þ β 0 ½u þ tan ϕðδ þ ξ ½uÞ 2 C  expð2 C½u=Gf Þ A closed-form solution thus can be obtained for ½u, when Φo ¼ 0:  1 0 0 1 a0 C 2 Gf @ @C exp a1 Gf A a0 C A ½ u ¼ Wo 2 a1 Gf a1 Gf C

(8.24)

(8.25)

with a0 ¼ α0 þ δ tanϕ and a1 ¼ β 0 þ ξ tanϕ and W0 is the main branch of the Lambert function W. An additional criterion must be considered to characterize tensile/extension fractures. In this case, the opening is made in the same direction and orientation than the normal vector of the crack, np ¼ n and ½u is the opening magnitude (see Fig. 8.2B). To do so, an anisotropic principal strain criterion with opening in mode I is used. It is better suited than a strength criterion because fractures in mode I are due to an extension at the level of walls as a result of the free surface created by the tunnel driving. The failure criterion ΦlI for these principal strain criteria can be written as: ΦlI ¼ supðnT Hs;T E 2 Ey ðnÞÞ

(8.26)

n

with Ey the maximum value of positive strain to stay in the linear domain. This value can depend on the normal vector n of the extension plane. After fracture initiation, n ¼ nI , the opening criterion ΦoI is calculated by: ΦoI ¼ nTI Hs;T E 2 Ey ðnI Þ  expð2 Ey ðnI Þ½u=LfI Þ

(8.27)

It is possible, in the same manner as the MohrCoulomb criterion to obtain with Lambert function an analytical solution of the crack opening in mode I and LfI is a material property, related to the fracture energy for mode I and elastic constants.

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8.4

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Application to 3D multicracking: induced fracture networks around drifts after an excavation in claystone

A FE model of cylindrical shape (length L ¼ 60 m and radius R ¼ 30 m), refined around the walls of a radius r ¼ 2:6 m excavation (Fig. 8.3) is used for numerical simulations with around 106 degrees of freedom (d.o.f.). The claystone is modeled as a transversely isotropic material with the following elastic properties: 8 ET ¼ Ex ¼ Ey ¼ 11; 000 MPa > > > > E < L ¼ Ez ¼ 5200 MPa G ¼ 2500 MPa > > ν > T ¼ 0:3; ν LT ¼ 0:2 > : ðν TL ¼ ν LT ET =EL Þ The anisotropy ratio about 2:0 corresponds to an upper limit from experimental campaigns. In situ stress field is as follows: 8 < σH ¼ σxx 5 2 16:2 MPa σ ¼ σyy 5 2 12:4 MPa : h σv ¼ σzz 5 2 12:7 MPa Moreover, the dissymmetrical fracture patterns observed around drifts, especially those excavated parallel to σH where the initial stress state at the drift section is quasi-isotropic demonstrates a possible anisotropy of strength. An anisotropic description is thus considered for both failure criteria. Regarding shear failure, the inherent anisotropy of the rock is taken into account through an anisotropic description of its cohesion. The cohesion depends on the orientation of the shear plane and the value is calibrated according to experimental data of compression strength for different loading orientations (Chen et al., 2012), C ¼ CðnÞ. A constant friction

Figure 8.3 Mesh,  106 d.o.f., L ¼ 60 m, R ¼ 30 m, r ¼ 2:6 m:

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angle is considered ðtanϕ ¼ 0:2Þ. Fig. 8.4 represents the cohesion as a function of the angle between the normal vector of the shear plane and the vertical axis. The fracture energy Gf value is 25 JUm22 . With elastic parameters of the claystone, it pffiffiffiffi corresponds to a fracture toughness about 0:6 MPa m. It is worth to note that there is not many data available on the fracture toughness of clay. The available data pffiffiffiffi show a rather weak fracture toughness ranging between 0.3 and 0.4 MPa m (Huang et al., 2014). However, a slightly greater value is used to limit calculated fracture openings. Regarding the principal strain criterion, the maximum value of strain Ey ðnÞ basis on which cracks in mode I are initiated also depend on the normal vector of the extension plane. Not many data is currently available to obtain this threshold from experimental results. Arbitrary values are thus chosen in manner to provide a weaker strength in lateral direction. This value is taken to 0:13% in the horizontal direction and 0:3% in the vertical direction with a linear progression between these two threshold values. Fig. 8.5 shows the cracking network around the walls for a drift parallel to the major horizontal stress σH . The excavated area is depicted in red. For both images on the left, 20m of rock is excavated and 30m for the two right images. The value of the sliding for fractures in mode II, initiated by the MohrCoulomb criterion and the value of crack opening for fractures with opening in mode I, initiated by the main extension criterion are represented. Numerical results show that cracks in mode I (about 60%) are outnumbered than cracks in mode II (about 40%). One of the main advantages of 3D modeling is the possibility to observe cracking upstream and downstream of the working face and to simulate a real excavation process. There is no need for using a deconfinement curve. The fracture network is complex, the representation in the form of several sections is better suited to understand and to observe the extent of the fractured zone. ! Fig. 8.6 shows cracks along the axis of the drift by a normal longitudinal section y ! for cracked elements in mode II and normal z for cracked elements in mode I. There are some side effects that we do not take into account. Most of crack opening/sliding values are on the order of a few millimeters. Among all cracked 6

θ

C (MPa)

5.5 5 4.5 4 Shearplane 3.5 3

0

20

40

60

θ(°)

Figure 8.4 Cohesion as a function of the orientation θ of the shear plane.

80

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Figure 8.5 Cracking (modes I and II) around the drift’s walls after an excavation process parallel to the major horizontal stress.

elements, some of them have an important value of opening or sliding and can be considered as a chipping. Then, for different sections, the fracture networks (elements cracked in mode I) are represented. The legend indicates the value of the radial displacement in millimeters. It can be seen that fractures are initiated before face arrival to a section and then propagate around the drift. The extent of fracture networks observed experimentally and numerically is similar for tensile fractures and we can find numerically the anisotropy of facies of cracking in mode I. We can observe that this anisotropy is increased with the distance between the considered section and the working face (Fig. 8.7). It is possible to quantify this anisotropy by plotting the energy dissipation (relative) in mode I as a function of the orientation α for a section situated at 15m

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Figure 8.6 Cracking around the drift’s walls, longitudinal sections parallel to the major horizontal stress.

Figure 8.7 Facies of cracking (in mode I) for different sections and radial displacement Ur in mm. Working face at 40m, excavation parallel to the major horizontal stress: facies at (A) 40.5 m, (B) 40_ m, (C) 38 m and (D) 25 m.

behind the working face (Fig. 8.8). The lateral extent of fractured zone is about twice its vertical extent. Fig. 8.9 shows the numerical value of convergence as a function of α for sections at different distance intervals from the working face. The convergence refers to the displacement between two diametrically opposite points of the tunnel wall from the initiation of the excavation process. For each interval, an average on

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Energy dissipation (relative)

1

0.8

0.6

0.4 α

0.2

0 0

30

60

90

120

150

180

α(°)

Figure 8.8 Anisotropy: energy dissipation (relative) in mode I as a function of the orientation α for a section situated at 15m behind the working face, excavation parallel to the major horizontal stress.

30 28

1–2 m 4–6 m 10–12 m 16–17 m

α

26

Convergence (mm)

24 22 20 18 16 14 12 10 0

30

60

90

120

150

180

α(°)

Figure 8.9 Convergence as a function of α at different distances from the face, excavation parallel to the major horizontal stress.

several sections was performed. In situ observations show that for a drift along the major horizontal stress, the magnitude of convergence is maximized in the horizontal direction with a ratio between horizontal and vertical convergence around two (Guayaca´n-Carrillo et al., 2015). Numerical simulations are in agreement with these observations. By standing away from the working face, this ratio is fairly constant regardless of the considered section and is worth about 1:5 (Fig. 8.9). The value of

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the vertical convergence is around 18mm far from the face and around 25mm for the horizontal convergence. It is in the same order of magnitude compared with in situ measurements (Guayaca´n-Carrillo et al., 2015). Convergence of walls is primarily due to cracking, and more particularly to cracking with opening in mode I (“tensile fractures”). Indeed, for a linear case, the radial displacement is of the order of a few millimeters so that it is of the order of several centimeters for a nonlinear problem (with fractures). In addition, the elastic anisotropy of the material explains that for a linear problem the horizontal convergence is lower than the vertical convergence (also because absolute value of σv is slightly higher than σh ) while cracking causes the reverse phenomenon (Figs. 8.7 and 8.9). It also explains why in the vicinity of the working face (12 and 46 m on Fig. 8.9) the horizontal convergence is lower than the vertical convergence. There is thus a competition between elastic anisotropy and fracturing process regarding the tunnel convergence. Orientations and the connectivity of shear fractures are shown in Fig. 8.10 for different sections. The connectivity of cracks is

Figure 8.10 Orientations of shear planes for different sections. Working face at 40m, excavation parallel to the major horizontal stress: facies at (A) 42 m, (B) 40 m, (C) 38 m, and (D) 30 m.

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important only near the walls. There is a lack of extension in mode II in numerical simulations but orientations of shear planes are correct. In situ observations show that the extent of shear fractures has a larger size than numerical results in the horizontal direction. Tensile fractures are located only near the wall. It appears as two different zones of excavation induced fracturing from in situ tests. The first consists on tensile/extension and shear fractures near the walls. A second one consists only on shear fractures with an anisotropic extent farther from walls. This second zone is missing in numerical simulations despite a slight anisotropy. Numerical results are in agreement with in situ observations for the position of cracks in mode I. Numerical parameters were adjusted with experimental data and differences should be interpreted with caution. In situ observations show a lateral pore pressure increase before the face arrival. This pore pressure evolution induces a slight anisotropy in effective stress field at the section of the tunnel. It is probably necessary to address these hydromechanical couplings to better represent the complex process of cracking around drifts which can not yet be correctly explained to date.

8.5

Conclusions

Among several techniques to represent brittle or quasi-brittle cracking within an FE model, we show that the strong discontinuity approach presents some decisive advantages: (1) the ability to use constitutive laws with softening without mesh dependency, (2) the representation of a cracks as a real discontinuity within the displacement field, (3) the computation of both cracks openings and their orientations. We also stress on the non intrusive nature of the E-FEM implementation for those “strong discontinuities.” Basically, dealing with the case of extensive multicracking, the E-FEM leads to a constant number of unknowns, whatever the number of cracks is. Moreover, this local implementation can be done within any FE code, providing that one can access to the “constitutive law” routine. Finally, we present an application of the E-FEM framework to a kinematics corresponding to sliding in 3D. This model is based on the well-known MohrCoulomb criterion. We show the simulation of an excavation process in claystone. The in situ observations show a very rich and complex cracking pattern close to the walls and we compare some of the measurements made to the 3D simulation.

References Aliabadi, M.H., 1997. A new generation of boundary element methods in fracture mechanics. Nucl. Eng. Design. 86, 91125. Armero, F., Garikipati, K., 1996. An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. Int. J. Solids Struct. 33 (2022), 28632885.

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Bai, Y., Wierzbicki, T., 2009. Application of extended MohrCoulomb criterion to ductile fracture. Int. J. Fract. 161, 120. Bazant, Z., Belytschko, T.B., 1985. Wave propagation in strain-softening bar: exact solution. J. Eng. Mech. 111, 381389. Belytschko, T., Fish, J., Engelmann, B.E., 1988. A finite element with embedded localization zones. Comput. Methods Appl. Mech. Eng. 70 (1), 5989. Brancherie, D., Ibrahimbegovic, A., 2009. Novel isotropic continuum-discrete damage model capable of representing localized failure of massive structures. Eng. Comput. 26, 100127. Charlotte, M., Laverne, J., Marigo, J.J., 2006. Initiation of cracks with cohesive force models: a variational approach. Eur. J. Mechanics A/Solids. 25, 649669. Chen, L., Shao, J.F., Zhu, Q.Z., Duveau, G., 2012. Induced anisotropic damage and plasticity in initially anisotropic sedimentary rocks. Int. J. Rock Mech. Mining Sci. 51, 1323. Dvorkin, E.N., Cuitino, A.M., Gioia, G., 1990. Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int. J. Numer. Methods Eng. 30 (3), 541564. Guayaca´n-Carrillo, L.M., Sulem, J., Seyedi, D.M., Ghabezloo, S., Noiret, A., Armand, G., 2015. Analysis of long-term anisotropic convergence in drifts excavated in CallovoOxfordian claystone. Rock Mech. Rock Eng. Hauseux, P., Roubin, E., Seyedi, D.M., Colliat, J.B., 2016. Fe modelling with strong discontinuities for 3d tensile and shear fractures: application to underground excavation. Comput. Meth. Appl. Mech. Eng. 309, 269287. Hill, R., 1958. A general theory of uniqueness and stability in elasticplastic solids. J. Mech. Phys. Solids. 6, 236249. Huang, Y., Xie, S.Y., Shao, J.F., 2014. An experimental study of crack growth in claystones. Eur. J. Environ. Civil Eng. 18, 307314. Jira´sek, M., 2000. Modeling of localized inelastic deformation. Comput. Methods Appl. Mech. Eng. 188, 307330. Jirsek, M. (2012). Modeling of localized inelastic deformation. Kanninen, M.F., Brust, F.W., Ahmad, J., Abou-Sayed, I.S., 1982. The numerical simulation of crack growth in weld-induced residual stress fields. In: Kula, E., Weiss, V. (Eds.), Residual Stress and Stress Relaxation. Plenum Press, New York, NY, pp. 975986. Larsson, R., Runesson, K., Sture, S., 1996. Embedded localization band in undrained soil based on regularized strong discontinuity theory and FE-analysis. Int. J. Solids Struct. 33 (2022), 30813101. Liaw, B.M., Kobayashi, A.S., Emery, A.F., 1984. Double nodding technique for mixed mode crack propagation studies. Int. J. Numer. Methods Eng. 20, 967977. Lloberas-Valls, O., Huespe, A., Oliver, J., Dias, I., 2016. Strain injection techniques in dynamic fracture modeling. Comput. Methods Appl. Mech. Eng. 308, 499534. Lotfi, H.R., Shing, P.B., 1995. Embedded representation of fracture in concrete with mixed finite elements. Int. J. Numer. Methods Eng. 38 (8), 13071325. Matthies, H., Strang, G., 1979. ). The solution of nonlinear finite element equations. Int. J. Numer. Methods Eng. 14 (11), 16131626. Mazars, J., Pijaudier-Cabot, G., 1989. Continuum damage theory-application to concrete. Journal of Engineering Mechanics. 115 (2), 345365. Moe¨s, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131150. Oliver, J., 1996a. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals. Int. J. Num. Meth. Eng. 39 (21), 35753600.

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Oliver, J., 1996b. Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals. Part 2: Numerical simulation. Int. J. Num. Meth. Eng. 39 (21), 35753623. Oliver, J., 2000. On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations. Int. J. Solids Struct. 7 (4850), 72077229. Oliver, J., Huespe, A.E., Sa´nchez, P.J., 2006. A comparative study on finite elements for capturing strong discontinuities: E-FEM vs x-FEM. Comput. Methods Appl. Mech. Eng. 195 (3740), 47324752. Oliver, J., Dias, I., Huespe, A., 2014. Crack-path field and strain-injection techniques in computational modeling of propagating material failure. Comput. Methods Appl. Mech. Eng. 274, 289348. Oliver, J., Caicedo, M., Roubin, E., Huespe, A., Hernndez, J., 2015. Continuum approach to computational multiscale modeling of propagating fracture. Comput. Methods Appl. Mech. Eng. 294, 384427. Ortiz, M., Leroy, Y., Needleman, A., 1987. A finite element method for localized failure analysis. Comput. Methods Appl. Mech. Eng. 61 (2), 189214. Peerlings, R.H., De Borst, R., Brekelmans, W., De Vree, J., Spee, I., 1996. Some observations on localisation in non-local and gradient damage models. Eur. J. Mech. A Solids. 15 (6), 937953. Pietruszczak, S., Mro´z, Z., 1981. Finite element analysis of deformation of strain-softening materials. Int. J. Numer. Methods Eng. 17, 327334. Pijaudier-Cabot, G., Bazant, Z., 1987. Nonlocal damage theory. J. Eng. Mech. 113, 15121533. Rashid, M.M., 1998. The arbitrary local mesh replacement method: an alternative to remeshing for crack propagation analysis. Comput. Methods Appl. Mech. Eng. 154, 133150. Roubin, E., Vallade, A., Benkemoun, N., Colliat, J.-B., 2015. Multi-scale failure of heterogeneous materials: a double kinematics enhancement for embedded finite element method. Int. J. Solids Struct. 52, 180196. Rudnicki, J.W., Rice, J.R., 1975. Conditions for the localization of deformation in pressuresensitive dilatant materials. J. Mech. Phys. Solids. 23, 371394. Seyedi, M., Taheri, S., Hild, F., 2006. Numerical modeling of crack propagation and shielding effects in a striping network. Nucl. Eng. Design. 236, 954964. Simo, J.C., Oliver, J., 1994. A new approach to the analysis and simulation of strain softening in solids. Fractude and Damage in Quasibrittle Structures. E & FN Spon, pp. 2539. (London, z. p. bazant, z. bittar, m. jirasek and j. mazars edition). Simo, J.C., Rifai, M., 1990. A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Numer. Methods Eng. 29, 15951638. Simo, J.C., Oliver, J., Armero, F., 1993. An analysis of strong discontinuities induced by strain-softening in rate independent inelastic solids. Comput. Mech. 12, 277296. Sluys, L.J., Berends, A.H., 1998. Discontinuous failure analysis for mode-I and mode-II localization problems. Int. J. Solids Struct. 35 (3132), 42574274. Washizu, K., 1982. Variational methods in elasticity and plasticity. third ed. Pergamon Press, New York, NY. Wells, G.N., Sluys, L.J., 2001. Three-dimensional embedded discontinuity model for brittle fracture. Int. J. Solids Struct. 38, 897913. Wilson, E.L., 1974. The static condensation algorithm. Int. J. Numer. Methods Eng. 8, 198203.

Application of continuum damage mechanics in hydraulic fracturing simulations

9

Amir K. Shojaei1 and Jianfu Shao2 1 DuPont Performance Materials, DuPont, Wilmington, DE, United States, 2Laboratory of Mechanics of Lille, Villeneuve d’Ascq, France

9.1

Introduction

Hydraulic fracturing (HF) is an evolving technology that has been contributing to the oil and gas production around the globe in past few decades. Stimulation treatments, such as HF, are crucial to achieve economic production from low permeability formations or reservoirs with low conductivity from natural fractures. A typical HF job includes three basic stages (1) fracture initiation, (2) fracture propagation, and (3) flow-back. Fig. 9.1 shows typical surface pressures during injection stages in a HF treatment. Depending on rock’s properties, fluid viscosity, flow rate, temperature, confining pressure, and others different HF scenarios are possible, as depicted by different lines in Fig. 9.1. Current practices in assessing and modeling HF treatments rely on simplified fracture and fluid flow models, which could only provide approximate solutions regarding the actual fracture geometry. Common assumptions in simulation of HF include homogeneous formation properties and limited fracture growth in a symmetric double-wing fashion; although these assumptions are erroneous in most of the unconventional reservoirs where geology, formation properties, and environmental conditions (such as high pressure high temperature) can complicate the fracturing process (Shojaei et al., 2014). In this chapter, the application of simulation techniques for HF treatment design is discussed in detail. The continuum damage mechanics (CDM) tool is introduced as a powerful and reliable analysis platform. The basis for the optimization cycle of HF design is also elaborated in which simulation techniques are utilized to study the effects of several design parameters on the cost and efficiency of HF jobs. In Section 9.5, several HF treatments are numerically studied to demonstrate the capabilities of the proposed CDM framework. It is shown that the CDM platform provides realistic simulations, and it may be utilized by design engineers to investigate HF treatments before field implementations. This chapter is structured as follows: In Section 9.2, the design procedure of HF is briefly discussed. The thermodynamics consistency of CDM models are discussed in Section 9.3. In Section 9.4, the CDM model formulation is presented. Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00009-9 © 2017 Elsevier Ltd. All rights reserved.

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Figure 9.1 Schematics for pressure versus time scenarios in HF. Scenarios 1, 2, and 3 indicate the effect of various parameters such as fluid viscosity, injection rate, formation mechanical properties, presence of natural fractures, and confining pressures on the pressure profile.

The simulation results and remarks are presented in Section 9.5. Both indices and bolded letters are used to denote tensorial parameters, and light letters indicate scalar parameters.

9.2

Simulation techniques and hydraulic fracturing design

The fracture height, orientation, and length in a HF job can be affected by sundry factors, including (1) crack closure stress differences between targeted formation and overlaying and underlying formations that results in unexpected fracture propagation to adjacent formations. (2) Boundary layers breakdown where shallow formation thicknesses of layered permeable and impermeable formations affect the HF crack geometry; (3) breakdown pressures, (4) modulus contrast between targeted and adjacent formations, (5) bedding plane slip at shallow depths, (6) natural fractures, (7) interface slip where fracture propagates between the boundary layers instead of formation, (8) injection fluid properties, such as viscosity, and (9) perforation characteristics such as perforation shape, depth, and orientation. One of the main objectives of HF design is to contain the fracture within the targeted formation. Fig. 9.2 shows the case that a nonperforated formation is fractured that may result in violation from original design requirements. Such unexpected scenarios may results in contamination of surface waters due to hydrocarbon migrations, and it also results in excessive operational costs. Thus, it is crucial to investigate the HF treatment before field implementation via lab scale testing and simulation techniques. Simulation techniques have been widely used to study the HF propagation (Papadopoulos et al., 1983; Shlyapobersky and Chudnovsky, 1992; Bai and Lin, 2014). Simulation techniques are essential tools for HF design optimization in which several design parameters, including geotechnical, perforation geometry,

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199

Figure 9.2 Schematics of designed (desired) and actual fractures.

injection rate, and fracturing fluid viscosity, are integrated into the simulation steps to predict the fracture height and depth in various formation configurations. Following pioneering works by Griffith (1921), most of the modeling efforts in HF are historically performed using linear elastic fracture mechanics (LEFM) in which an ideal crack is simulated in an ideal formation (Perkins and Kern, 1961; Geertsma and De Klerk, 1969; Ghassemi et al., 1998). Although such simplified analytical solutions can provide some insight into the HF treatment; there are strong evidence that fracture toughness could be a function of confining pressure and the length scale. Fracture propagation analysis in LEFM models requires prescribing the fracture path in which parameters, for example, the energy release rate and stress intensity factors, controls the fracture initiation and propagation. Many of those parameters are unknown for the downhole conditions, and the prediction of the fracture length, velocity, and orientation is not reliable in many practical cases. Three-dimensional (3D) LEFM models have also been developed such as cellbased, lumped, and coupled fluid flow models (Adachi et al., 2007, 2010). Cohesive zone models (CZM) are intended to address shortcomes of LEFM approach, in which cohesive elements together with force
displacement relations are required to model the fracture (Chen et al., 2009; Sarris and Papanastasiou, 2011, 2012). Although CZM can predict fracture initiation and avoid singularity at tip of the crack, it encounters with a few computational difficulties such as magnitude of cohesive parameters depending on the mesh size, and a random and tortuous crack path needs to be replaced with a line that undermines the prediction results. Extended finite-element method (XFEM) has also been widely used to simulate fracture process in HF processes (Sukumar et al., 2000). Although calibration of the material parameters for XFEM models is complicated, both CZM and XFEM methods require a specific type of element to be used in fracture process zone to simulate the fracture that results in higher computational costs. Boundary element method (BEM) provides an approach for solving steady-state flow in 3D fracture networks (Wilson and Witherspoon, 1974; Elsworth, 1987). Although BEM reduces the model dimension and reduces the computational cost, the solution may not be as accurate as finite-element analysis (FEA) in dealing with heterogeneous rock layers. Simulation of crack growth using the FEA has also been widely practiced in the literature (Advani et al., 1990; Moe¨s et al., 1999). Mesh dependency of the

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results together with high computational cost are the main disadvantages of the fracture mechanics based FEA methods. Most recently, CDM of porous rocks under HF is being introduced as a powerful and reliable simulation technique. In this approach, full coupling between poroelasticity, poroplasticity, and fracture mechanics of rocks are integrated into the simulation steps to provide realistic results (Shao, 1998; Shao et al., 2004; Shojaei et al., 2014). In this approach, prescription of the fracture paths is not necessary, and the fractures evolve naturally on the basis of poroelasticity, poroplasticity, and CDM constitutive laws of the rock system. The CDM model can be defined on the basis of strain energy terms, stress, strain, or a combination of these parameters. The main aims of a CDM formulation are outlined as follows: 1. Full coupling between poroelasticity, poroplasticity, and damage mechanisms of rocks is considered to provide realistic simulations of fracture in porous rocks. 2. Different deformation mechanisms, including poro-elasto-plastic and damage mechanics, can be calibrated with respect to specific experimental data. In contrast to many fracture simulation techniques, CDM approach provides a clear workflow to link experimental data to model parameters. 3. The pore pressure driven fractures are simulated by CDM. In other words, CDM model links the fluid pressure to damage mechanics of rocks in a hydraulic fracture simulation. So a fluid pressure driven fracture is modeled by CDM, unlike LEFM models.

9.3

The drained and/or undrained conditions can be studied

1. Implementation of a CDM model in an FEA solver provides the design engineers with a design evaluation platform. This design platform can then be utilized to numerically study the effect of sundry factors, for example, heterogonous formation layups, anisotropic rock system, porosity effect, permeability effect, presence of natural fractures, temperature dependency, and loading rate dependency, on the HF shape, depth, and orientation. 2. CDM models utilize the conventional solid elements that eliminate the needs for specialized elements, or remeshing techniques, and consequently may reduce the complexity of the computational implementation.

As mentioned before, CDM models can be calibrated with respect to the available experimental data that ensure the simulation data correlating well with the real fracturing process at downhole conditions.

9.4

Thermodynamic principles and continuum damage mechanics

Thermodynamic principles, including energy considerations, can be utilized to derive physically consistent constitutive laws for the material behavior. The deformation process in porous rocks can be categorized into reversible, for example, poroelastic response, and irreversible processes, for example, damage mechanisms and

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201

poroplastic behaviors. In the context of thermodynamic of solids, the specific internal energy u is defined as follows (Voyiadjis et al., 2011): u 5 uðs; Ee ; ζ p ; ζ d Þ;

(9.1)

where s denotes the entropy, Ee ; ζ p ; and ζ d are respectively elastic strain, plastic, and damage variable tensors. Time derivative of u reads u_ 5

@u @u @u _ @u _ s_ 1 :_E e 1 :ζ p 1 :ζ @s @Ee @ζ p @ζ d d:

(9.2)

The symbol ‘‘:’’ in the above equation indicates contraction over two tensorial indices. The second law of thermodynamic states that the change in entropy is always positive, and it can be expressed in the Clausius
Duhem inequality as follows (Voyiadjis et al., 2011):
 q σ:_E e 2 ρ u_ 1 sT_ 2 :rT $ 0; T

(9.3)

where ρ denotes the density of the rock. By substituting Eq. (9.2) into Eq. (9.3) and eliminating the heat flux term, the Clausius
Duhem inequality for the rock system is obtained as follows: !     @u @u @u _ @u _ σ2ρ :ζ 1 :ζ $ 0: s_ 2 ρ :_E e 2 ρ T 2 (9.4) @Ee @s @ζ p p @ζ d d The conjugate thermodynamic forces, namely Cauchy stress tensor σ, and temperature T, respectively corresponding to the entropy s and the elastic strain tensor Ee are defined as follows: σ5ρ

@u ; @Ee

and T 5

@u : @s

(9.5)

The remaining part of Eq. (9.4) define the plasticity and damage power of dissipation, ! @u _ @u _ Γ5 2ρ :ζ 1 :ζ : (9.6) @ζ p p @ζ d d Helmholtz-free energy function Ψ is obtained through Legendre transformation of the internal energy u (Voyiadjis et al., 2011). The Helmholtz-free energy potential can be decomposed into (1) purely elastic, and (2) coupled plastic and damage, Ψpd ; parts (Hansen and Schreyer, 1994), Ψ5



 1
E 2 Ep :CðdÞ: E 2 Ep 1 Ψpd d; Ep ; 2

(9.7)

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where d is the damage tensor. In the case of isotropic materials, following Hill’s notation, the effective elastic stiffness tensor of damaged material CðdÞ is given by (Nemat-Nasser and Hori, 1993), CðdÞ 5 2μðdÞK 1 3kðdÞJ;

(9.8)

where kðdÞ and μðdÞ represent the bulk and shear moduli of the damaged rock system, respectively. The two isotropic symmetric fourth-order tensors J and K are defined by (Shao, 1998). The thermodynamic conjugate forces for each flux variables, Ee ; Ep , and d, are obtained as follows (Shao, 1998; Voyiadjis and Shojaei, 2015a,b):
 @Ψ 5 CðdÞ : E 2 Ep ; @Ee @Ψ 5 σ; γp 5 2 ρ @Ep

σ52ρ

0

1

 @Ψ d;E @Ψ 1 @C ð d Þ pd p A: 5 2 ρ @ E 2 Ep : : E 2 Ep 1 γd 5 2 ρ @d 2 @d @d

(9.9)




Both plasticity and damage mechanisms require definition of criteria that control initiation of each processes (Voyiadjis and Shojaei, 2015a,b),
 f p 5 γp 2 σy Ep # 0; f d 5 γd 2 r ðdÞ # 0;

(9.10)

where σy and r control the initiation of plasticity and damage processes, respectively. The principal of extremized entropy production during a thermodynamic process is applied to the power dissipation function Eq. (9.6) considering the constraints conditions given in Eq. (9.10). The resulting Lagrangian functional that should be extremized is as follows (Voyiadjis et al., 2011): p d γ  5 Γ 2 λ_ f p 2 λ_ f d

(9.11)

p d where λ_ and λ_ are Lagrangian constant, to be found from extermized functional. Applying the stationary conditions to the Lagrangian functional results in (Voyiadjis et al., 2011), p @γ  p @f 5 0 ! E_ p 5 λ_ ; @σ @σ d @γ  _ 5 λ_ d @f : 5 0 ! d @γ d @γ d

(9.12)

Eq. (9.8) represents thermodynamic consistent constitutive laws for the plasticity and damage phenomena.

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9.4.1 Continuum damage mechanics of porous rocks CDM approaches can be formulated to simulate complex failure mechanisms in porous rocks under high confining pressures at elevated temperature. The fracture simulation relies on hydraulic-driven fracture where injected pore pressure is utilized to determine the fracture initiation and propagation. The basics building blocks of CDM computational tools consist of G

G

G

G

Mathematical models for pressure and temperature sensitive rock’s properties. A CDM model that captures the initiation and propagation of microscale failure mechanisms. Poroelasticity and poroplasticity constitutive relations are required to calculate the stress
strain fields due to applied hydromechanical, and thermal loads. For complex geometries, such as multilayer rocks with multiple natural fractures, an FEA solver is needed for numerical computations. An explicit or implicit integration algorithm can be used to solve the numerical problem.

Damage mechanics in rocks is a multiscale phenomenon that starts from microcracks/microvoids and eventually results in rupture of the structure upon formation of macroscale/structural scale cracks, see Fig. 9.3. CDM provides a tool to simulate the initiation and propagation of microcracks, formation of macroscale crack by coalescence of microflaws, and eventually simulation of final rupture. The damage mechanics was formerly introduced by pioneers such as Kachanov (1958) and Lemaitre (1984), and since then it has been under intensive research and developments (Voyiadjis and Kattan, 2006; Voyiadjis et al., 2011; Voyiadjis and Shojaei, 2015a,b). Effective and damaged configurations in CDM basically capture the continuum damage processes in which damaged materials are removed from the material, and the load is only carried by the undamaged material. As depicted in Fig. 9.4, the reduction in material stiffness is then embedded in the numerical simulations to model material softening due to the damage

Figure 9.3 A schematic representation of multiscale damage mechanisms in different material systems.

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Figure 9.4 Effective and damaged configurations that are used to develop continuous reduction of material stiffness by the damage parameter d, after [17].

mechanisms (Voyiadjis et al., 2012). Damage parameter d correlates the undamaged stiffness E to the damaged stiffness E through two basic principles of (1) equivalence of the strain energy, or (2) equivalence of the strain, refer to Voyiadjis et al. (2012) for more details. To address the fracture in porous rocks, the role of pore pressure in damage mechanics needs to be integrated into the analysis. Shojaei et al. (2014) utilize Biot’s theory of consolidation in anisotropic rock to incorporate pore pressure driven fracture into the simulation. The total strain rate E_ is decomposed into elastic E_ e , plastic E_ p , and damaged E_ d strain rate tensors, E_ 5 E_ e 1 E_ p 1 E_ d :

(9.13)

Then, Biot’s theory of consolidation (Detournay and Cheng, 1993; Cheng, 1997) that utilizes coupled diffusion and elasticity equations to capture stress fields in porous media is incorporated, 0

σ 5 M h E 2 αp

(9.14)

Application of continuum damage mechanics in hydraulic fracturing simulations

205

0

where σ is the so-called effective stress tensor, α is the Biot coefficients tensor, and M h is the fourth-order homogenized stiffness tensor. One may note that unlike the isotropic case, the Biot coefficient is not scalar. The relation between pore pressure p, strain field E and Biot’s coefficients ζ, M; and α is given by (Detournay and Cheng, 1993; Cheng, 1997; Shojaei et al., 2014), p 5 M ðζ 2 αEÞ:

(9.15)

It is worth mentioning that the Biot’s coefficient together with stiffness tensor will be updated on the basis of the damage parameter with the course of deformation. Pressure-sensitive poroplasticity for drained/undrained conditions are simulated through a set of simple empirical relations (Shojaei et al., 2014). The shear strength of the rock’s solid skeleton τ is related to pressure sensitive ultimate strength τ u via two material parameters B and n as follows (Shojaei et al., 2014):  n τ 5 τ u 1 B  Ep 

(9.16)

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Ep  5 2=3Ep : Ep is the equivalent plastic strain. The parameter τ u is then correlated to the hydrostatic pressure field via two equations. First is a linear correlation that defines τ u based on the tensile hydrostatic of the rock sample T, and τ u0 0 which is the strength at effective confining pressure Σ0 . The second relation applies 0 0 for the Σ . Σ0 (Shojaei et al., 2014),


0 0  τ u 5 τ u0 1 τ umax 2 τ u0 1 2 exp 2 α Σ 2 Σ0

(9.17)

0

where Σ 5 Σ 2 p is the effective confining pressure, and τ umax is the maximum strength of the rock that occurs at high confining pressures, and α is defined by (Shojaei et al., 2014), α5


τ umax

2 τ u0

τu : 0
0 3 Σ0 1 T

(9.18)

The final step is to formulate the microcracks, and microvoids initiation and propagation within the solid skeleton of the porous rock. Fig. 9.5 shows the damage mechanics in porous rocks in which microcracks/microvoids degrade the elastic properties of the solid skeleton. The actual damage process in porous rocks is a 3D process that results in anisotropic material properties after damage initiation. Thus, even for an initially isotropic rock sample, the state of material properties becomes anisotropic due to anisotropy in the applied stress field. Anisotropic CDM formulation, developed by Shojaei et al. (2014), reads

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Figure 9.5 Representative element for (A) undeformed porous rock under pressure and shear loading, (B) unloaded medium that contains microvoids and microcracks.

2

D1 d54 0 0

0 D2 0

3 0 0 5 D3

(9.19)

where Di , with i 5 1; 2; and 3 are three damage parameters in principal directions that is updated with the course of deformation. The damage parameter is then correlated to the microcrack lengths as follows (Shojaei et al., 2014): Di 5

a i 2 a0 ac 2 a0

(9.20)

where ai is a representative of all microcracks at the ith principal direction, ac is the critical, and a0 is the initial microcrack length in the system. The microcrack evolution law is then given by (Shojaei et al., 2014) ai 5 a0 1 χ5 ðac 2 a0 Þ

X jEp j Ef

(9.21)

where χ5 is a material parameter that controls the rate of microcracking, and Ef is the fracture strain. The stiffness of the porous rocks is updated via the damage parameters as follows (Shojaei et al., 2014):

 2 0:5 Ei 5E ð12Di Þ; and ν i 5 v δi 1Di 21 v

(9.22)

where E and v are intact elastic modulus and Poisson’s ratio, respectively. A fracture mechanics based damage initiation criterion is also used by Shojaei et al. (2014) to minimize the mesh sensitivity of the results that is associated with highly localized softening due to the damage mechanisms. The CDM criterion then reads (Shojaei et al., 2014)

Application of continuum damage mechanics in hydraulic fracturing simulations



GI fD 5 GIc

207

2              GII 2 GIII 2 GI GII GII GIII GI GIII 1 1 1 1 1 GIIc GIIIc GIc GIIc GIIc GIIIc GIc GIIIc (9.23)

where Gi (i 5 I, II, and III) is the fracture energy release rate for modes I, II, and III crack openings that is computed numerically, and Gic (i 5 I, II, and III) is the critical fracture energy release rates for the respective fracture modes (Shojaei et al., 2014).

9.5

Simulation results

The performance of the developed CDM model is compared to the laboratory scale testing in which brown-colored sandstone experimental data, after Khazraei (1995), are used to calibrate the plasticity and damage models (Shojaei et al., 2014). The poro-elasto-plastic deformation response of the rock system was experimentally investigated by Khazraei (1995) through triaxial tests. As depicted in Figs. 9.6 and 9.7, the developed plasticity model by Shojaei et al. (2014) correlates well with the experimental data. A computer-aided design (CAD) model of a rock sample is depicted in Fig. 9.8 (Shojaei et al., 2014). Three domains are considered for the problem—low-permeability cap rocks on the top and bottom of the model, a hydrocarbon-bearing zone, and interfacial domains to represent potential fracture paths (Fig. 9.8). The oilbearing circular rock slice has a 30-m depth and 400-m radius with a wellbore radius of 0.1 m. An orthotropic overburden stress state is imposed. Loading and boundary conditions are also applied as follows. The first step is initially achieving

Figure 9.6 Pressure-sensitive stress
strain responses of porous rocks (Khazraei, 1995; Shojaei et al., 2014).

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Porous Rock Failure Mechanics

Figure 9.7 Damage response in limestone under quasistatic loading condition, after Shojaei et al. (2014).

Figure 9.8 CAD model for simulation of fracture in a targeted rock sample (Shojaei et al., 2014).

the equilibrium state after applying the initial pore pressure and in-situ stresses. The second step simulates the HF stage, where a volume of fluid is injected into the formation. The fluid flow is injected along the perforation zone in the target formation in the model by means of the prescribed interfacial medium (Fig. 9.9). The interfacial domain, with thickness of 0.1 m, shows the fracture process zone in which the damage parameters are updated and elastic moduli degradation occurs. The elasto-plastic rock behavior is assigned to the target rock, whereas confining shale rocks are assumed to behave as linear elastic materials. The duration of the injection stage is 140 s. Following the HF, another transient consolidation analysis is conducted. The injection into the well is terminated, and fluid leakoff from the fracture is allowed to bleed off the fracture fluid pressure. To mimic actual conditions, the fracture surfaces are assumed to possess a minimum opening as the boundary condition to simulate the behavior of the placed proppant material into the fracture (Shojaei et al., 2014). It is worthwhile to note that two types of elements are used in Fig. 9.9—C3D8R for oil-bearing rock and C3D8RP for the interfacial medium in which the pore fluid

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injection process can be simulated (Shojaei et al., 2014). In the next case study, the interaction between HF and natural fractures is studied. Fig. 9.10 shows the pore pressure distribution in different stages of the injections, starting from initiation to propagation stages. The fracture tip can be traced by following the pore pressure distribution in Fig. 9.10. In Fig. 9.11, the von Mises stress distribution is depicted in three stages of the injection process. Again the tip of the HF crack is traceable, and the speed of the crack propagation can be computed from the model. Fig. 9.11 shows the damage propagation when a natural fracture intersects a HF fracture during the treatment process. The natural fracture is assumed to be made of the same rock properties of the HF medium. The natural fracture intersects with the HF path at a 90-degree angle. Due to flexibility of the CDM approach, different material properties and shapes can be readily assigned to the natural fracture to study the influence of presence of natural fractures on the HF growth.

Figure 9.9 Pore pressure variations within the interfacial rock medium (fracture process zone) (Shojaei et al., 2014).

Figure 9.10 Simulation results for the von Mises stress in which the travel of the fracture tip along the fracture process zone is shown.

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Figure 9.11 Effective damage parameter during treatment process at 30, 40, and 100 s of injection from left edge are shown. The natural fracture intersects the HF fracture at a 90-degree angle.

9.6

Concluding remarks

Basic principles for designing a successful HF treatment is elaborated in this chapter in which several design parameters are introduced for HF design optimization tasks including (1) number/geometry of perforations, (2) fracturing fluid properties for example, viscosity, (3) proppant concentration, (4) injection rates, and others. The design optimization of HF via simulation techniques is then discussed in detail in this chapter. Various numerical schemes are outlined, and their pros and cons are briefly described. A robust numerical tool, based upon CDM and poroplastic deformation of rocks, is introduced to predict the growth of hydraulic fractures within targeted formations. Anisotropic poroplasticity is used to bridge the pore pressure effect to the state of stress, strain, and damage in porous rocks. It is assumed that excessive plastic deformation in the skeleton of the porous rocks results in microcracks, microvoids nucleation, and propagation. An anisotropic CDM model is developed that correlates the rate of microcracking to the thermo-mechanical loads and pore pressure. The CDM model is developed on the basis of the fracture mechanics of porous rocks, and it takes into account the microcrack and microvoid nucleation and propagation within the solid skeleton of the porous rocks. The main contributions of the discussed CDM framework are as follows: G

G

G

A standalone or built-in software can be developed to accurately predict/simulate the growth of HF in various formations and layup configurations. Developed software would provide a powerful design tool for HF optimization tasks where several uncertainty seniors can be studied simultaneously. The HF simulation can be carried out realistically in which poroelasticity, poroplasticity, and fracture mechanics of porous rocks are utilized to calibrate the proposed CDM framework.

In order to optimize the HF design, assessment of porous rocks’ fracture mechanics at downhole condition is essential. The main concern in terms of HF simulation is the correlation between simulation and actual hydraulic fractures at

Application of continuum damage mechanics in hydraulic fracturing simulations

211

downhole conditions. Currently, the authors are developing new testing methods for experimentally simulating the downhole conditions. Such testing techniques are crucial for CDM model verification and validation steps.

References Adachi, J., Siebrits, E., Peirce, A., Desroches, J., 2007. Computer simulation of hydraulic fractures. Int. J. Rock Mech. Min. Sci. 44 (5), 739
757. Adachi, J.I., Detournay, E., Peirce, A.P., 2010. Analysis of the classical pseudo-3D model for hydraulic fracture with equilibrium height growth across stress barriers. Int. J. Rock Mech. Min. Sci. 47 (4), 625
639. Advani, S.H., Lee, T.S., Moon, H., 1990. Energy Considerations Associated With the Mechanics of Hydraulic Fracture. Society of Petroleum Engineers, Columbus, OH. Bai, J., Lin, A., 2014. Tightly Coupled Fluid-Structure Interaction Computational Algorithm for Hydraulic Fracturing Simulations. American Rock Mechanics Association, Minneapolis, MN. Chen, Z., Bunger, A.P., Zhang, X., Jeffrey, R.G., 2009. Cohesive zone finite element-based modeling of hydraulic fractures. Acta Mech. Solida Sin. 22 (5), 443
452. Cheng, A.H.D., 1997. Material coefficients of anisotropic poroelasticity. Int. J. Rock Mech. Min. Sci. 34 (2), 199
205. Detournay, E., Cheng, A.H.-D., 1993. Fundamentals of poroelasticity. In: Hudson, J.A. (Ed.), Comprehensive Rock Engineering: Principles, Practices and Projects, Vol. 2. Pergamon Press, Oxford, UK. Elsworth, D., 1987. A boundary element-finite element procedure for porous and fractured media flow. Water Resour. Res. 23 (4), 551
560. Geertsma, J., De Klerk, F., 1969. A rapid method of predicting width and extent of hydraulically induced fractures. Journal o f Petroleum Technology. 21 (12), 1571
1581. Ghassemi, A., Diek, A., Roegiers, J.C., 1998. A solution for stress distribution around an inclined borehole in shale. Int. J. Rock Mech. Min. Sci. 35 (4
5), 538
540. Griffith, A.A., 1921. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. A, Containing Pap. Math. Phys. Character. 221 (582
593), 163
198. Hansen, N.R., Schreyer, H.L., 1994. A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Struct. 31 (3), 359
389. Kachanov, L.M., 1958. Rupture time under creep conditions. Izvestija Academii Nauk SSSR. 8, 26
31 (Reprinted in International Journal of Fracture, 97, 11
18). Khazraei, R. (1995). Experimental Investigations and Numerical Modelling of the Anisotropic Damage of a Vosges Sandstone., University of Lille. Doctoral Thesis. Lemaitre, J., 1984. How to use damage mechanics. Nucl. Eng. Des. 80 (2), 233
245. Moe¨s, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. Int. J. Numerical Methods Eng. 46 (1), 131
150. Nemat-Nasser, S., Hori, M., 1993. Micromechanics: Overall Properties of Heterogeneous Materials. Elsevier., Amsterdam. Papadopoulos, J.M., Narendran, V.M., Cleary, M.P., 1983. Laboratory Simulations of Hydraulic Fracturing. Society of Petroleum Engineers, Denver, CO. Perkins, T.K., Kern, L.R., 1961. Widths of hydraulic fractures. Journal of Petroleum Technology. 13 (9), 937
949.

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Sarris, E., Papanastasiou, P., 2011. The influence of the cohesive process zone in hydraulic fracturing modelling. Int. J. Fract. 167, 33
45. Sarris, E., Papanastasiou, P., 2012. Modeling of hydraulic fracturing in a poroelastic cohesive formation. Int. J. Geomech. 12 (2), 160
167. Shao, J.F., 1998. Poroelastic behaviour of brittle rock materials with anisotropic damage. Mech. Mater. 30 (1), 41
53. Shao, J.F., Lu, Y.F., Lydzba, D., 2004. Damage modeling of saturated rocks in drained and undrained conditions. J. Eng. Mech. 130, 733
740. Shlyapobersky, J., Chudnovsky, A., 1992. Fracture Mechanics in Hydraulic Fracturing. American Rock Mechanics Association, Santa Fe, NM. Shojaei, A., Dahi Taleghani, A., Li, G., 2014. A continuum damage failure model for hydraulic fracturing of porous rocks. Int. J. Plast. 59 (0), 199
212. Sukumar, N., Moe¨s, N., Moran, B., Belytschko, T., 2000. Extended finite element method for three-dimensional crack modelling. Int. J. Numerical Methods Eng. 48 (11), 1549
1570. Voyiadjis, G.Z., Shojaei, A., 2015a. Continuum Damage-Healing Mechanics Healing mechanics. In: Voyiadjis, Z.G. (Ed.), Handbook of Damage Mechanics: Nano to Macro Scale for Materials and Structures. Springer New York, New York, NY, pp. 1515
1539. Voyiadjis, G.Z., Shojaei, A., 2015b. Thermodynamics of Continuum Damage Healing Mechanics Healing mechanics. In: Voyiadjis, Z.G. (Ed.), Handbook of Damage Mechanics: Nano to Macro Scale for Materials and Structures. Springer New York, New York, NY, pp. 1493
1513. Voyiadjis, G.Z., Shojaei, A., Li, G., 2011. A thermodynamic consistent damage and healing model for self healing materials. Int. J. Plast. 27 (7), 1025
1044. Voyiadjis, G.Z., Shojaei, A., Li, G., Kattan, P.I., 2012. A theory of anisotropic healing and damage mechanics of materials. Proc. R. Soc. A: Math. Phys. Eng. Sci. 468 (2137), 163
183. Voyiadjis, Z., Kattan, P.I., 2006. Advances in Damage Mechanics. Elsevier, London. Wilson, C.R., Witherspoon, P.A., 1974. Steady state flow in rigid networks of fractures. Water Resour. Res. 10 (2), 328
335.

Multiscale modeling approaches and micromechanics of porous rocks

10

Wanqing Shen and Jianfu Shao Lille University of Science and Technology, Villeneuve d’Ascq, France

10.1

Introduction

Most rocks are heterogeneous materials and their macroscopic behaviors are inherently related to their microstructures such as mineral compositions and porosity. Based on extensive experimental evidences and within the framework of irreversible thermodynamics, many phenomenological plastic and damage models have been developed. These models are generally able to capture various features of mechanical behaviors of rock-like materials, but they are not providing explicit relationships between macroscopic responses and microstructures. In order to develop an alternative modeling method and to improve phenomenological models, significant efforts have been undertaken during the last decades on the development of micromechanical models. In particular, different analytical and numerical homogenization techniques for composite materials have been successfully adapted for rock-like materials. Different micromechanical models have then formulated for the description of crack-induced damage and plastic deformation. It is not the objective of this chapter to give a comprehensive review of all micromechanical models so far developed for rock-like materials. As an example, we propose to present here a micromechanical model for modeling elastic-plastic behavior of porous rocks with two families of pores at two different scales. The kind of microstructure can be representative for a wide class of rocks, for instance clayey rocks, chalk, etc. It is possible to identify intraparticle pores inside mineral grains (plates) and interparticle pores between mineral grains. These two families of pores affect in a different way the macroscopic properties of material such as elastic modulus, plastic yield stress, and failure strength (Homand and Shao, 2000; Papamichos et al., 1997; Schroeder, 2003; De Gennaro et al., 2004; Alam et al., 2010; Xie and Shao, 2006). Further, due to the presence of pores, two plastic deformation processes can be generally identified, the plastic pore collapse under mean effective stress and the plastic shearing under deviatoric stress. The objective of micromechanical approach is to establish a macroscopic plastic criterion explicitly taking into account the effects of pores. As a pioneer work, Gurson (1977) proposed an analytical yield criterion using a kinematical limit analysis approach, for porous metal materials constituted of a pressure-independent Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00010-5 © 2017 Elsevier Ltd. All rights reserved.

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von Mises type solid containing a spherical void. A large number of extended criteria have been developed by various authors for different kinds of engineering materials including rocks like materials. For example, considering a Drucker-Prager type pressure-sensitive solid matrix with spherical pores, different macroscopic plastic yield criteria have been formulated for instance (Jeong, 2002; Guo et al., 2008; Lee and Oung, 2000; Durban et al., 2010; Shen et al., 2015). On the other hand, the voids shape effects have been studied by Gologanu et al. (1997), Pardoen and Hutchinson (2003), Monchiet et al. (2014). More recently, Vincent et al. (2009) formulated a semianalytical macroscopic yield criterion for porous materials with two populations of pores and a von Mises solid phase at the microscopic scale. The same material has been studied in Shen et al. (2012a). However, it is found that the obtained elliptic plastic yield surface did not fit well a number of experimental data observed in laboratory tests (Homand and Shao, 2000; Papamichos et al., 1997; Schroeder, 2003; Xie and Shao, 2006). A new closed-form macroscopic criterion has been derived by Shen et al. (2014) for porous materials with a Drucker
Prager type solid phase and two populations of pores using a limit-analysis approach. In this chapter, the principle of the limit analysis method is first shortly recalled. Using this method and based on the previous work of Shen et al. (2014), the formulation of the macroscopic yield criterion for porous rocks is presented. After introducing a nonassociated plastic potential and plastic hardening function, the micromechanics-based plastic model is formulated and then applied to describe the mechanical behaviors of a typical chalk.

10.2

Principle of the limit analysis method

In this section, we first recall some basic principles of the limit analysis method for porous materials. In order to derive an explicit expression of macroscopic yield criterion, a hollow sphere Ω with an internal radius a and an external one b is usually chosen as the studied representative volume element (RVE) to consider the influence of porosity f 5 a3 =b3 ; or a cell of a porous material made up of a spheroidal volume Ω containing a confocal spheroidal void ω to account for the effects of pore shape and porosity simultaneously. The RVE is subjected to a homogeneous strain rate D on the outer surface: vðxÞ 5 D  x;

’ xA@Ω

(10.1)

The matrix in the RVE obeys to a local convex yield function FðσÞ # 0. The microstress tensor field is statically admissible and plastically admissible. With an associated flow rule, the microscopic strain rate d in the matrix can be calculated: _ @FðσÞ ; d5Λ @σ

_ ΛFðσÞ 5 0;

FðσÞ # 0;

_ $0 Λ

(10.2)

_ is the local plastic multiplier. Associated to the velocity field v, the strain where Λ rate tensor d is kinematically admissible: d 5 ðgrad v1 t grad vÞ=2.

Multiscale modeling approaches and micromechanics of porous rocks

217

According to Salenc¸on (1990), the corresponding support function can be expressed πðdÞ 5 supfσ:djFðσÞ # 0. For the uniform strain rate boundary conditions considered in the present study, the following inequality holds for all macroscopic stress Σ and macroscopic strain rate D (Hill
Mandel lemma): X

 :D # ΠðDÞ 5 inf vKA

1 jΩj



ð Ω2ω

πðdÞdV

(10.3)

ΠðDÞ represents the macroscopic dissipation, The infimum in (10.3) is taken over all kinematically admissible velocity fields, v. In the framework of limit analysis, the choice of the velocity field v(x) in the matrix complying with the boundary condition (10.1) is one of the key points to derive the macroscopic yield criterion. As classically, the limit stress states of the plastic porous medium at the macroscopic scale are shown to be of the form: X

10.3

5

@Π @D

(10.4)

Macroscopic criterion of double porous rock

In the framework of limit analysis method, an explicit macroscopic criterion will be derived in this section for a class of porous rock with two populations of voids at different scales. A two-step homogenization procedure will be developed. the RVE of the studied double porous material is defined in Fig. 10.1. For the sake of simplicity, we assume that both families of pores are spherical and randomly distributed in a Inter-particle pore (Large pore) Ω2 Solid phase

Ωm Equivalent homogeneous material

Ω

Macroscopic Scale Total porosity f (1–φ) + φ

(A)

r a

Intra-particle pore (Small pore) Ω1

Porous matrix

b

Mesoscopic Scale Ω Meso porosity φ = Ω2

(B)

Microscopic Scale Ω Micro porosity f = Ω –1 Ω

2

(C)

Figure 10.1 The RVE of the studied double porous medium at different scales.

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Porous Rock Failure Mechanics

solid matrix. At the macroscopic scale, the studied material can be seen as an equivalent homogeneous material (see Fig. 10.1A). The interparticle pores (large pores) of the studied porous medium are found at the mesoscopic scale. The matrix in Fig. 10.1B itself is a porous medium that is composed of intraparticle pores (small pores) and the solid phase at the microscopic scale (Fig. 10.1C). The two populations of spherical voids are distributed at two well-separated scales. We denote jΩj the total volume of the RVE, Ω2 the volume of the large voids at the mesoscopic scale, Ω1 and Ωm are the domains occupied by the small voids and the solid phase at the microscopic scale, respectively. With these notations, the porosity at the microscopic scale (intraparticle pores) f, the one at the mesoscopic scale φ (inter-particle pores), and the total porosity Γ at the macroscopic scale can be expressed as f5

jΩ1 j ; jΩ 2 Ω2 j

φ5

jΩ2 j ; jΩj

Γ5

jΩ1 j 1 jΩ2 j 5 f ð1 2 φÞ 1 φ jΩj

(10.5)

A two-step homogenization procedure will be adopted here to derive a macroscopic criterion for the studied material. In the first homogenization from microscale to mesoscale, the effects of microporosity f and the plastic compressibility of the solid phase is taken into account. In the second one from mesoscale to macroscale, a macroscopic plastic criterion will be obtained in the framework of limit analysis theory with considering the mesoporosity φ. The sign conventions of stress and strain are as follows: tensile stress (strain) is positive whereas compressive stress (strain) is negative. For the sake of clarity, the microscopic stress in the solid phase (Fig. 10.1C) is denoted as ~ the mesoscopic and macroscopic ones are σ and Σ, respectively. σ,

10.3.1 Homogenization from microscopic to mesoscopic scale For most geomaterials, the plastic behavior is generally affected by the mean stress and exhibits volumetric compressibility or dilatancy. In the first homogenization step from the microscopic scale to mesoscopic scale (see Fig. 10.1C), the plastic behavior of the solid phase is here assumed to obey to a Drucker
Prager criterion: ~ 5 σ~ d 1 Tðσ~ m 2 hÞ # 0 Φm ðσÞ

(10.6)

pffiffiffiffiffiffiffiffiffiffiffiffi ~ the mean stress, and σ~ d 5 σ~ 0 : σ~ 0 the equivalent deviatoric where σ~ m 5 trσ=3 stress with σ~ 0 5 σ~ 2 σ~ m 1. The parameter h represents the hydrostatic tensile strength, whereas T denotes the frictional coefficient. For such a porous medium, Maghous et al. (2009) have derived an effective strength criterion by using the modified secant method (interpreted by Suquet (1995), as equivalent to the variational method of Ponte Casta neda (1991)). F mp ðσ; f ; TÞ 5

  1 1 2f =3 2 3f σ 1 2 1 σ2m 1 2ð1 2 f Þhσm 2 ð12f Þ2 h2 5 0 d T2 2T 2 (10.7)

Multiscale modeling approaches and micromechanics of porous rocks

219

This criterion explicitly takes into account the effect of the porosity at the microscopic scale f. It also presents a tension
compression asymmetry that is a manifestation of the pressure sensitive solid phase. This criterion is then adopted here to describe the plastic behavior of the porous matrix.

10.3.2 Homogenization from mesoscale to macroscale During the second homogenization step from the mesoscopic to macroscopic scale, the macroscopic mechanical behavior of the double porous rock is determined by using a limit analysis approach. As shown in Fig. 10.1B, the porous rock is represented by a hollow sphere containing the porosity φ at the mesoscopic scale. Based on (10.7) and for a general case, the effective plastic criterion of the porous matrix can be written in the following general elliptic form: ΦðσÞ 5 βσ2eq 1

9α ðσm Þ2 2 Lσm 2 σ20 # 0 2

(10.8)

The scalars α, β, L, and σ0 are material constants that can be identified from Eq. (10.7) for the porous matrix. The representation of this elliptic criterion (10.7) or (10.8) corresponds to a closed surface. According to (10.8), the mesoscopic strain rate is given by the normality rule:     _ 3βσ0 1 3α σm 2 L 1 d5Λ 9α

(10.9)

where 1 is the second-order unit tensor. 2 5 ð2=3Þd0 :d0 and With the definition of the equivalent strain rate deq 2 0 0 _ is defined: Λ _ 5 deq =2βσeq . σeq 5 ð3=2Þσ :σ . The plastic multiplier Λ The plastic dissipation πðdÞ of the porous matrix can be calculated: πðdÞ 5 σ:d 5

 deq  Lσm 1 2σ20 2βσeq

(10.10)

By using the relationships of (10.9) and (10.8), the expressions of σeq and σm are given in the following form: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2 u σ0 1 L u 18α ; σeq 5 u 2 2 u 2β dm tβ 1 2 α deq

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 L2 u L 2β dm u σ0 1 18α u σm 5 1 2 2 9α 3α deq u tβ 1 2β dm 2 α deq

(10.11)

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Porous Rock Failure Mechanics

Substituting Eq. (10.11) into Eq. (10.10), the mesoscopic plastic dissipation πðdÞ finally readsa L dm 1 πðdÞ 5 3α

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 deq L2 2dm2 1 σ20 1 18α α β

(10.12)

According to (10.3), the macroscopic plastic dissipation ΠðDÞ can be obtained (Suquet, 1985; Buhan, 1986):  ΠðDÞ 5 inf vKA

1 jΩj



ð Ω2Ω2

πðdÞdV

(10.13)

For the moment, it is necessary to construct a kinematically admissible velocity fields v. Based on the one used by Gurson (1977), the following trial velocity field is adopted to account for the plastic compressibility of the matrix: v 5 Ax 1

b3 ðDm 2 AÞ e r 1 D0  x r2

(10.14)

in which Dm 5 ð1=3Þtr D and D0 the deviatoric part of the macroscopic strain rate D. The field Ax is homogeneous and allows us to account for matrix plastic compressibility. The two remaining terms are kinematically admissible with (D 2 A1). More precisely, the second term in Eq.(10.14) corresponds to the expansion of the cavity and the outer volume, whereas the third one describes the shape change of the cavity and of the outer boundary without volume change. Hence, for any value of the scalar A, the whole velocity field v complies with the uniform strain rate D applied to the hollow sphere. Due to the presence of A (which remains unknown in the definition of the velocity field), the macroscopic dissipation, ΠðDÞ is computed owing to a minimization procedure with respect to A:



~ ΠðDÞ 5 min ΠðD; AÞ ; A

~ ΠðD; AÞ 5

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 σ20 1 18α Ω

L vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ð 2 3α ð u2dm deq t 1 dV 1 dm dV Ω Ω2Ω2 α β Ω2Ω2

a

More details can be found in Shen et al. (2014).

(10.15)

Multiscale modeling approaches and micromechanics of porous rocks

221

The determination of the macroscopic criterion requires then to compute the integrals of deq and dm over the matrix. The strain rate in the matrix can be obtained from (10.14), in spherical coordinates, as

b3 ðDm 2 AÞ 1 2 3e r  e r 3 r

d 5 A1 1 D0 1

(10.16)

From which, one can get  dm 5 A;

2 deq 5 D2eq 1 4

2

b3 ðDm 2AÞ 4 b3 ðDm 2 AÞ 0 1 D : 1 2 3e r  e r 3 3 r 3 r (10.17)

In order to obtain a closed-form expression, the following inequality is classically used: ð

ð Ω2Ω2

deq dV 5

Ω2Ω2

ð b D E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 deq ðr; θ; ϕÞdV # 4π r 2 deq D

2 deq

dr

SðrÞ

a

where SðrÞ is the sphere of radius r and

1=2

E SðrÞ

(10.18)

2 is the average of deq ðr; θ; ϕÞ

over all the orientations: D

2 deq

E SðrÞ

5

1 4π

ð SðrÞ

2 deq ds 5 4

 3 2 b ðDm 2AÞ 1 D2eq r3

(10.19)

This eventually yields an upper bound of the macroscopic dissipation by computing: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð u ffi du u L2 1=φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L ~ ΠðD; AÞ 5 tσ20 1 Að1 2 φÞ M 2 1 N 2 u2 2 1 u 3α 18α 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" ffi #1=φ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u u 2 L2 uN L M 2 1N 2 u2 t 5 σ0 1 Að1 2 φÞ 1 2 N  arcsinh M 3α 18α u 1

(10.20)      where M 2 5 2A2 =α 1 D2eq =β , N 2 5 4=β ðDm 2AÞ2 and the change of variable u 5 b3 =r 3 has been introduced. ~ As mentioned before, one has to minimize ΠðD; AÞ over the unknown variable A and to determine the macroscopic yield function by taking advantage of the approx~ imate expression of ΠðD; AÞ. In practice, rather than treating these two steps

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Porous Rock Failure Mechanics

successively, the computation of the criterion may be done by addressing them simultaneously (cf. Monchiet, 2006). It comes X

5

~ @ΠðD; AÞ ; @D

~ @ΠðD; AÞ 50 @A

with

(10.21)

To solve (10.21), it is convenient as in Shen et al. (2012a) to make the following ~ ~ change of variable: ΠðD; P AÞ 5 ΠðM; NÞ with M and N introduced before. The macroscopic stress tensor then reads X

5

~ ~ @M ~ @N @Π @Π @Π 5 1 @D @M @D @N @D

(10.22)

The condition of minimization with respect to A (second relation in (10.21)) becomes ~ @M ~ @N @Π @Π L 1 1 ð1 2 φÞ 5 0 @M @A @N @A 3α

(10.23)

Similarly to the approach used by Gurson, one can then establish the parametric form of the macroscopic yield function: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u u u ~ @Π u 2 L2 4u N2 u 2 N2 5 t1 1 tφ 1 ΣA 5 5 tσ0 1 2 @M 18α M2 M2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "   #  u ~ u 2 @Π L2 N N 5 tσ 0 1 ΣB 5 2 arcsinh arcsinh @N φM M 18α

(10.24)

By eliminating the parameter N=M in the above two relationships, one gets 0

12

0

1

B C B C ΣA ΣB Brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 2φcoshBrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiC 2 1 2 φ2 5 0 @ A @ 2 2 A L L σ20 1 σ20 1 18α 18α

(10.25)

The macroscopic yield function (10.25) is of Gurson type with appropriate quantities ΣA and ΣB that needs to be explicit. To this end, noting that M depends only on the deviation D0 and scalar A, whereas N is a function of Dm and A, it comes P m

5

~ ~ 1 @Π 2 @Π ; 5 pffiffiffi 3 @Dm 3 β @N

X

0

5

~ ~ @Π 2D0 @Π 5 0 @D 3βM @M

(10.26)

Multiscale modeling approaches and micromechanics of porous rocks

223

in which pffiffiffi ~ @Π 3 β 5 Σm @N 2

(10.27)

~ Considering the condition of minimization with respect to A, (10.23), @Π=@M finally reads ~ @Π 5 @M

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 9α L Σm 2 ð12φÞ 1 βΣ2eq 2 9α

(10.28)

The closed-form expression of the macroscopic criterion of the porous medium having a matrix obeying to the general elliptic criterion (10.8) is 2 32 L  2  pffiffiffi  ð12φÞ Σ 2 m Σeq 9α 6 3 β Σm 7 9α β 1 2 1 2 φ2 5 0 4 5 1 2φ cosh Σ0 2 Σ0 2 Σ0 (10.29) with Σ0 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi σ20 1 L2 =18α . When L 5 0, α 5 0, and β 5 1, we obviously retrieve

the Gurson (1977) criterion. Let us come back to the problem of the double porous material with a Drucker
Prager (10.6) solid phase. The parameters β, α, L, and σ0 can be determined by comparing the two elliptic criteria (10.7) and (10.8). As a final result, the closed-form expression of the macroscopic criterion of the double porous material having a compressible solid phase at the microscale is derived: 32 2 2ð12f ÞhT 2   2  6Σm 1 3f 22T 2 ð12φÞ7 7 6 2 4f Σeq 3f 2 6 7 1 2T 6 Φ5 1 7 3 9 2 Σ0 Σ0 5 4

(10.30)

! sffiffiffiffiffiffiffiffiffiffiffi 3 Σm 1f 1 2φ cosh 2 1 2 φ2 5 0 2 Σ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Σ0 5 ð1 2 f ÞhT 3f =ð3f 2 2T 2 Þ This macroscopic criterion takes into account simultaneously the compressibility of the solid phase, the influences of the microporosity f and the meso one φ. Fig. 10.2 shows the influences of the two porosities f and φ on the yield surface that has a tension
compression asymmetry. The total macroscopic porosity is the same (43%) in these three cases, but different proportions of microporosity f and

224

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Figure 10.2 Yield surfaces predicted by (10.30) with different proportions of microporosity f and mesoporosity φ, h 5 32 MPa, T 5 0:3, total porosity Γ 5 0:43.

meso porosity φ are considered. The macroscopic yield surface of the studied double porous material is thus clearly affected by the proportion of f and φ, in particular for the compressive loading.

10.4

Formulation of a nonassociated elastoplastic model for double porous material

In this section, a complete micromechanics-based constitutive model is proposed for double porous materials. The elastic behavior is firstly considered. Due to the microstructure of the studied REV, the macroscopic elastic stiffness tensor ℂhom can be calculated by the classical upper bound of Hashin and Shtrikman (1963) for the two steps of homogenization procedure. Knowing the values of the bulk and shear modulus κs and μs of the solid phase at the microscopic scale and the intraparticle porosity f, the effective bulk and shear modulus (κp and μp ) of the porous matrix at the mesoscale (Fig. 10.1B) can first be calculated by κp 5

4ð1 2 f Þκs μs ; 4μs 1 3f κs

μp 5

ð1 2 f Þμs κs 1 2μs 1 1 6f 9κs 1 8μs

(10.31)

Multiscale modeling approaches and micromechanics of porous rocks

225

Figure 10.3 Influence of the porosity ratio f =φ on the effective elastic properties of double porous material with a total porosity Γ 5 0:43.

In the second homogenization step, the effective elastic properties of the double porous matrix can be determined by considering the influence of the interparticle porosity φ. Based on the result (10.31), the homogenized bulk and shear modulus κhom and μhom of the double porous medium are given by κhom 5

4ð1 2 φÞκp μp ; 4μp 1 3φκp

μhom 5

ð1 2 φÞμp κp 1 2μp 1 1 6φ 9κp 1 8μp

(10.32)

The influences of the mesoporosity φ and the microporosity f on the effective elastic properties are shown in Fig. 10.3. Young’s modulus and Poisson’s ratio of the solid phase are taken as Es 5 12 GPa and υs 5 0:2, respectively. The total porosity of the studied double porous material is Γ 5 0:43. The homogenized bulk and shear modulus κhom and μhom are affected by the ratio of f and φ. When the proportion of the microporosity f is low (the ratio f =φ is small), the values of κhom and μhom are big. In the domain 0 , f =φ , 5, the influence of this ratio on the effective elastic properties is especially important. For the elastoplastic behavior, the macroscopic plastic criterion (10.30) presented in Section 10.3 will be applied. Most geomaterials exhibit a plastic hardening. In the present study, this effect on the macroscopic behavior is taken into account via the evolution of the frictional coefficient T as a function of the equivalent plastic strain εpeq in the solid phase: Tðεpeq Þ.

226

Porous Rock Failure Mechanics

In most rock-like materials, it is generally necessary to develop a nonassociated plastic flow rule to more accurately describe plastic volumetric deformation. Inspired by the work of Maghous et al. (2009), Shen et al. (2012b, 2013), the following function is proposed as the macroscopic plastic potential by taking a similar form as the yield function (10.30) and introducing the dilatancy coefficient t: 2

32 2ð12f ÞhT 2 6Σm 1 3f 22Tt ð12φÞ7   2  6 7 2 4f Σeq 3f 7 1 2 Tt 6 1 G5 6 7 Σ0 3 9 2 Σ0 4 5

(10.33)

0sffiffiffiffiffiffiffiffiffiffiffi 1 3 Σm A 1f 1 2φ cosh@ 2 Σ0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with Σ0 5 ð1 2 f ÞhT 3f =ð3f 2 2TtÞ. The dilatancy coefficient t controls the transition between volumetric contractancy and dilatancy during plastic deformation. The macroscopic plastic potential depends also on the two porosities (f and φ) and the properties of the solid phase (T, t). According to the macroscopic potential (10.33), the plastic flow rule is given by @G ðΣ; f ; φ; T; tÞ Dp 5 λ_ @Σ

(10.34)

The equivalent plastic strain of the solid phase can be computed: ε_ peq 5

Σ:Dp  ð1 2 f Þð1 2 φÞ Th 1 ðt 2 TÞ

Σm ð1 2 f Þð1 2 φÞ



(10.35)

The evolutions of microporosity f and mesoporosity φ are respectively evaluated

  by f_ 5 d=dt Ω1 =ðΩ1 1 Ωm Þ and φ_ 5 d=dt Ω2 =Ω : _1 _m1Ω _m Ω Ω 2 ð1 2 f Þ f_ 5 ð1 2 f Þ Ωm 1 Ω1 Ωm _11Ω _2 _1 _m 1Ω _m 1Ω Ω Ω 2 ð1 2 φÞ φ_ 5 ð1 2 φÞ Ωm 1 Ω1 1 Ω2 Ωm 1 Ω1

(10.36)

With the increment of strain ΔE, the corresponding stress ΔΣ can be computed by using the macroscopic elastic stiffness tensor ℂhom : ΔΣ 5 ℂhom : ðΔE 2 ΔEp Þ;

ℂhom 5 3κhom J 1 2μhom K

(10.37)

Multiscale modeling approaches and micromechanics of porous rocks

227

  in which K 5 I 2 J, I is the fourth-order unity tensor and J 5 1=3 1  1. ΔEp is the plastic part of the incremental strain ΔE, which can be calculated by the macroscopic potential (10.33): ΔEp 5 dλ

@G @Σ

(10.38)

where the plastic multiplier dλ can be obtained from the consistency condition: @Φ _ @Φ _ @Φ _ _ 1 _ ðΣ; f ; φ; T Þ 5 @Φ :Σ f1 T 50 Φ φ1 @Σ @f @φ @T

(10.39)

Substituting (10.35), (10.36) and (10.37) into (10.39), one has: 

 @Φ :ℂ:ΔE @Σ dλ5 3 2 @G    Σ: @Φ @G @Φ @G @Φ @T 7 @Φ 6 @Σ :ℂ: 2 ð12f Þ4dv2t R 2dv 2 5 2 ð12φÞ ð12f Þð12φÞTh1ðt2TÞΣm @Σ @Σ @f @φ @Σm @T @εp

(10.40) in which   3f 2Σm @G 1 2ð1 2 φÞh Σ:@G 2 1 Σ: 2T 2 12φ @Σ @Σ : R5 ; dv 5 ~m Σm 12φ ð1 2 f Þð1 2 φÞTh 1 ðt 2 TÞΣ 2 2 2ð12φÞ h 2 2ð1 2 φÞh 12φ

Then the proposed micro
macro constitutive model is implemented in a standard finite element code (Abaqus) via a subroutine UMAT. Fig. 10.4 illustrates the stress
strain curves predicted by this model in the case of associated flow rule without plastic hardening. For the solid phase, Young’s modulus Es 5 12 GPa and Poisson’s ratio υs 5 0:2, h 5 32 MPa, t0 5 T0 5 0:3. The effects of proportions of microporosity f and mesoporosity φ on the overall strength of the double porous material with a total porosity Γ 5 0:43 can be obviously seen, for uniaxial and triaxial compression tests.

10.5

Application to a typical porous rock

The proposed model is applied in this section to describe the mechanical behavior of the porous chalk with a nonassociated plastic flow rule. The studied so-called Lixhe chalk is from the Upper Campanian age and was drilled in the CBR quarry near Lie`ge (Belgium). It is composed of more than 98% of CaCO3 and less than

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Porous Rock Failure Mechanics

(A)

(B) Σ11 (MPa)

Σ11–Σ33 (MPa)

–7,5

–7,5

–6,5

–6,5

–5,5 –4,5 –3,5 –2,5

–5,5

f = 0.1, φ =0.3667 f = 0.2, φ =0.2875

–4,5

f = 0.3, φ =0.1857 f = 0.4, φ =0.0500 Total porosity Γ =0.43

–1,5

1,5

E11 (%)

–0,5

E 33 0,5

–0,5

–1,5

–2,5

–3,5

f = 0.1, φ =0.3667 f = 0.2, φ =0.2875

–2,5

f = 0.3, φ =0.1857 f = 0.4, φ =0.0500

–1,5

Total porosity Γ =0.43

E11 (%)

–0,5

E 33 1,5

–3,5

0,5

–0,5

–1,5

–2,5

–3,5

Figure 10.4 Stress—strain curves predicted by the proposed model under compression test with different proportions of f and φ, total porosity Γ 5 0:43. (A) Uniaxial compression test, (B) triaxial compression test (5 MPa).

0.8% of SiO2 and 0.15% of Al2 O3 . The average porosity is 43%. According to Talukdar et al. (2004) and Alam et al. (2010), two populations of pores can be found in porous chalks: the microporosity f 5 40% and the meso one φ 5 5%. Based on works of Homand and Shao (2000) and Xie and Shao (2006), the following isotropic plastic hardening law is adopted:   p  m T 5 T0 1 1 b εpeq 1 c enεeq 2 1

(10.41)

in which T0 is the initial plastic yield threshold of the solid phase. b, m, c, and n are hardening parameters, which can be determined from hydrostatic compression tests. εpeq is the equivalent plastic deformation in the solid phase at the microscopic scale. As the rate of volumetric dilatancy generally varies with plastic deformation history, it is assumed that the dilatancy coefficient is also a function of the equivalent plastic strain in the solid phase. For the simplicity, the same form as (10.41) is adopted for the evolution of t:   p  m t 5 t0 1 1 b εpeq 1 c enεeq 2 1

(10.42)

in which the values of the parameters b, m, c, and n are the same as the ones used in Eq. (10.41).

10.5.1 Identification of model’s parameters Before applying the micro
macro model to describe the macroscopic behavior of the Lixhe chalk, both the elastic and plastic parameters of the proposed model are identified. The macroscopic elastic properties (Young’s modulus E, Poisson’s ratio υ, or the bulk and shear modulus κhom , μhom , respectively) are obtained from the

Multiscale modeling approaches and micromechanics of porous rocks

–40

229

Hydrostatic stress (MPa)

–35 –30 –25 –20 Experiment

–15 Simulation

–10 –5

Volumetric strain (%) 0 –0,5

–1,5

–2,5

–3,5

–4,5

–5,5

–6,5

Figure 10.5 Simulation of a hydrostatic compression test of chalk.

Table 10.1

Typical values of parameters for the non-associated

model Young’s modulus

Poisson’s ratio

Frictional coefficient

Dilatancy coefficient

Hydrostatic tensile strength

Plastic hardening

Es

vs

T0

t0

h

b

m

c

n

12 GPa

0.2

0.2

0.16

32 MPa

1.3

0.03

0.018

38

initial linear part of stress
strain curve during a conventional triaxial compression test. Then the modulus (κp , μp ) of the porous matrix at the mesoscale and the ones (κs , μs ) of the solid phase at the microscale can calculated by inverting Eqs. (10.32) and (10.31), respectively. The plastic parameters are then determined from the numerical fitting of a hydrostatic compression test (see Fig. 10.5). The typical values obtained are given in Table 10.1. We can see that the two plastic strain stages are correctly described in Fig. 10.5 by the model. In the first stage, a high plastic strain rate is obtained due to important plastic collapse of interparticle pores. In the second stage, due to the progressive increase of contacts between particles and also to the decrease of two porosities, the plastic strain rate is decreasing like the consolidation mechanism in granular materials. Further, the good agreement between the numerical results and experimental data indicates that the model’s parameters are well identified.

10.5.2 Experimental validation of micro
macro model The proposed nonassociated micro
macro model is now used to describe the mechanical behavior of the Lixhe chalk with the values given in Table 10.1. Using the same set

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Porous Rock Failure Mechanics

(B)

(A) –20 –18

–25

Σ11–Σ33(MPa)

–16

E33

Σ11–Σ33(MPa)

–20

E11

–14

E33

–12

E11

–15

–10 –10

–8 Experiment

–6

Experiment

Simulation

–4

Simulation

–5

–2 (%)

0 3

1

–1

–3

(%)

0

–7 2

–5

1

0

–1

–2

–3

–4

–5

–6

Figure 10.6 Comparison of stress
strain curves between numerical results and experimental data—triaxial compression test on Lixhe chalk with different confining pressures. (A) 14 MPa, (B) 17 MPa.

(B)

(A) 0,41

0,0505

f

0,4

14 MPa 17 MPa

0,39

φ

0,05

14 MPa 17 MPa

0,0495

0,38

0,049

0,37

0,0485

0,36

0,048 0,0475

0,35 E11

0,34

0

–0,02

–0,04

–0,06

E11

0,047

0

–0,02

–0,04

–0,06

Figure 10.7 Evolution of f and φ predicted by the proposed model with different confining pressures: (A) microporosity f, (B) mesoporosity φ.

of parameters, numerical simulations are then performed for triaxial compression tests with different confining pressures in order to verify the capacity of the proposed model. Fig. 10.6 illustrates the comparisons of stress
strain curves between experimental data and numerical results with 14 and 17 MPa confining pressures. Generally, an overall good agreement is observed. Both the axial and lateral strains are well predicted by the proposed model. The proposed model is able to capture the main features of the mechanical behaviors of Lixhe chalk, such as the dependence of confining pressure, effect of inter-particle and intra-particle pores, compressibility of the solid phase. Fig. 10.7 shows the evolutions of microporosity f in the matrix and of mesoporosity φ predicted by the proposed model as functions of axial strain E11 during triaxial compression tests with different confining pressures (14 and 17 MPa). It is found that the evolutions of f and φ are different. The changes of the microstructure affect the macroscopic plastic criterion (10.30) and the potential (10.33) that control the macroscopic behavior. With the decrease of the porosity, the plastic yield

Multiscale modeling approaches and micromechanics of porous rocks

231

surface expand and the strength of the studied chalk increases. The mechanical behavior of the Lixhe chalk is clearly sensitive to these two families of porosities. This is the main difference between the micromechanics based model and the phenomenological one.

10.6

Concluding remark

In this chapter, as an example of multiscale approaches, we have presented a micromechanical elastic-plastic model for a class of porous rocks with two populations of pores and a pressure sensitive solid matrix. Using a two-step homogenization procedure and a limit analysis method, an analytical macroscopic yield criterion has been established. This criterion explicitly takes into account the different effects of two populations of pores at different scales. Completed with a plastic potential and hardening law, a micromechanical elastic-plastic model has been formulated. This model can be as easily as any phenomenological models implemented into a standard computer code and then applied to engineering applications. The micro
macro model has been applied to describe the macroscopic behavior of a typical porous rock, the Lixhe chalk. It was found that the mean features of the mechanical behavior of the studied material were correctly described by the micromechanical model. In the future work, the micromechanical model can be extended to saturated and partially saturated porous materials by considering effects of fluid pressure in two populations of pores.

References Alam, M.M., Borre, M., Fabricius, I., Hedegaard, K., Røgen, B., Hossain, Z., et al., 2010. Biot’s coefficient as an indicator of strength and porosity reduction: calcareous sediments from Kerguelen plateau. J. Petrol. Sci. Eng. 70, 282
297. Buhan, P.D., 1986. A fundamental approach to the yield design of reinforced soil structures— chap. 2: yield design homogenization theory for periodic media. The`se d’e´tat. Universite´ Pierre et Marie Curie, Paris VI, France. De Gennaro, V., Delage, P., Priol, G., Collin, F., Cui, Y., 2004. On the collapse behaviour of oil reservoir chalk. Ge´otechnique. 54 (6), 415
420. Durban, D., Cohen, T., Hollander, Y., 2010. Plastic response of porous solids with pressure sensitive matrix. Mech. Res. Commun. 37, 636
641. Gologanu, M., Leblond, J., Perrin, G., Devaux, J., 1997. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.), Continuum Micromechanics. Springer Verlag. Guo, T., Faleskog, J., Shih, C., 2008. Continuum modeling of a porous solid with pressure sensitive dilatant matrix. J. Mech. Phys. Solids. 56, 2188
2212. Gurson, A., 1977. Continuum theory of ductile rupture by void nucleation and growth: part1yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol. 99, 2
15. Hashin, Z., Shtrikman, S., 1963. A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids. 11, 127
140. Homand, S., Shao, J., 2000. Mechanical behavior of a porous chalk and water/chalk interaction. Part I: Experimental study. Oil Gas Sci. Technol. 55 (6), 591
598.

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Jeong, H., 2002. A new yield function and a hydrostatic stress-controlled model for porous solids with pressure-sensitive matrices. Int. J. Solids Struct. 39, 1385
1403. Lee, J., Oung, J., 2000. Yield functions and flow rules for porous pressure-dependent strainhardening polymeric materials. J. Appl. Mech. 67, 288
297. Maghous, S., Dormieux, L., Barthe´le´my, J., 2009. Micromechanical approach to the strength properties of frictional geomaterials. Eur. J. Mech. A/Solid. 28, 179
188. Monchiet, V., 2006. Contributions a` la mode´lisation microme´canique de l’endommagement et de la fatigue des me´taux ductiles. The`se. Universite´ des sciences et Technologies de Lille, France. Monchiet, V., Charkaluk, E., Kondo, D., 2014. Macroscopic yield criteria for ductile materials containing spheroidal voids: an Eshelby-like velocity fields approach. Mech. Mater. 72, 1
18. Papamichos, E., Brignoli, M., Santarelli, F., 1997. An experimental and theoretical study partially saturated collapsible rock. Int. J. Mech. Cohesive-Frictional Mater. 2, 251
278. Pardoen, T., Hutchinson, J., 2003. Micromechanics-based model for trends in toughness of ductile metals. Acta Mater. 51, 133
148. Ponte Castan˜eda, P., 1991. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids. 39, 45
71. Salenc¸on, J., 1990. An introduction to the yield theory and its applications to soil mechanics. Eur. J. Mech. A/Solids. 9 (5), 477
500. Schroeder, C., 2003. Du coccolithe au re´servoir pe´trolier; approche phe´nome´nologique du comportement me´canique de la craie en vue de sa mode´lisation a` diffe´rentes e´chelles. Phd, thesis. University of Lie`ge. Shen, W.Q., Shao, J.F., Dormieux, L., Kondo, D., 2012a. Approximate criteria for ductile porous materials having a Green type matrix: application to double porous media. Comput. Mater. Sci. 62, 189
194. Shen, W.Q., Shao, J.F., Kondo, D., Gatmiri, B., 2012b. A micro-macro model for clayey rocks with a plastic compressible porous matrix. Int. J. Plast. 36, 64
85. Shen, W.Q., Kondo, D., Dormieux, L., Shao, J.F., 2013. A closed-form three scale model for ductile rocks with a plastically compressible porous matrix. Mech. Mater. 59, 73
86. Shen, W.Q., He, Z., Dormieux, L., Kondo, D., 2014. Effective strength of saturated double porous media with a Drucker
Prager solid phase. Int. J. Numer. Anal. Meth. Geomech. 38, 281
296. Shen, W.Q., Shao, J.F., Kondo, D., De Saxce, G., 2015. A new macroscopic criterion of porous materials with a Mises-Schleicher compressible matrix. Eur. J. Mech. A/Solids. 49, 531
538. Suquet, P., 1985. Homogenization techniques for composite media. In: Sanchez-Palencia, E. (Ed.), Ch. Elements of Homogenization for Inelastic Solid Mechanics. Springer Verlag, pp. 193
278. Suquet, P., 1995. Overall properties of nonlinear composites: a modified secant moduli approach and its link with Ponte Castan˜eda’s nonlinear variational procedure. R.R. Acad. Sci. Paris, IIb. 320, 563
571. Talukdar, M.S., Torsaeter, O., Howard, J., 2004. Stochastic reconstruction of chalk samples containing vuggy porosity using a conditional simulated annealing technique. Transport Porous Media. 57, 1
15. Vincent, P.-G., Monerie, Y., Suquet, P., 2009. Porous materials with two populations of voids under internal pressure: I. Instantaneous constitutive relations. Int. J. Solids Struct. 46, 480
506. Xie, S., Shao, J., 2006. Elastoplastic deformation of a porous rock and water interaction. Int. J. Plast. 22, 2195
2225.

Dynamic fracture mechanics in rocks with application to drilling and perforation

11

Amir K. Shojaei1 and George Z. Voyiadjis2 1 DuPont Performance Materials, DuPont, Wilmington, DE, United States, 2Louisiana State University, Baton Rouge, LA, United States

11.1

Introduction

Continuum damage mechanics (CDM) provides a continuum level description for microflaws initiation, propagation, and their coalescence that eventually results in large-scale macroscopic fractures and faults. To describe progressive degradation mechanisms many micromechanics based CDM models have been proposed based on experimental studies. CDM models may combine the theory of fracture mechanics with a statistic treatment to account for the random distribution of microcracks to study the dynamic damage evolution in brittle materials (Fahrenthold, 1991; Yang et al., 1996; Yazdchi et al., 1996; Wang et al., 2013). There is a wealth of literature on experimental and theoretical rock fracture mechanics and only a few of the most pertinent works are reviewed herein. Wang et al. (2013) studied the initiation and propagation of fractures around a preexisting cavities in brittle rocks. Yang et al. (2009) studied dynamic fracture of rocks under different impact rates in which three-point bending beam samples were subjected to different impact loads. Wu et al. (2015) developed a cohesive fracture model to simulate rock dynamic fractures in terms of mineral and cement properties. Gui et al. (2016) utilized a cohesive fracture model that combines tension, compression and shear material responses to simulate fracturing process in rock dynamic tests. Lattice bond cell models are utilized by Zhang et al. (2015) to account for the characteristics of the mesostructure of rock in which wave propagation in rock samples were investigated. Liu et al. (2015) proposed a dynamic damage constitutive model for jointed rock mass and it is applied to the rock mass models with many persistent joints. The intrinsic length scale and strain rate effects on the strength of rock-like materials were investigated by Qi et al. (2016). Dynamic Mode-I fracture parameters for rocks were determined using cracked chevron notched semi-circular bend specimen (Dai et al., 2011). In the context of the dynamic continuum mechanics, the prescription of the boundary and initial conditions completely define the problem. The microfracture processes, for example, microcracking or microvoiding, which introduce new Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00011-7 © 2017 Elsevier Ltd. All rights reserved.

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boundaries to the body, violate the continuum assumption of the medium. To regain the complete definiteness of the dynamic problem, additional information about the speed and trajectory of the crack propagation are required (Slepyan, 2010). The crack growth in the brittle and ductile solids typically reveals complex patterns of crack branching (Xu and Needleman, 1994). However, a full field analysis for fast crack propagation, in which cracks can initiate and propagate freely, in dynamic problems as well as quasi-static problems is lacking in the literature. Only, a few predictive techniques have been developed to simulate specific cases. Among those one can cite the dynamic crack growth models, which are based on the maximum energy dissipation, (Slepyan, 2010), wavy crack models that incorporate deviation of the crack growths from their plane limits (Cox et al., 2005), and cohesive zone models, in which the potential surfaces for decohesion are predicted based on energy descriptions (Xu and Needleman, 1994). In the case of cohesive zone models, a material length scale is incorporated to characterize the separation between cohesive surfaces, and the material failure modes that are embedded into the constitutive characterization of the cohesive surfaces. The CDM approach provides an alternative framework by which to describe the microscale damage process in materials. In this approach, a continuous description of the elastic properties degradation controls the microdamage processes. Most of the up-to-date CDM models rely on uniaxial test data to calibrate and characterize their material parameters. Application of such models in multiaxial loading scenarios is always questionable. In order to close the gap between uniaxially calibrated fracture models in multiaxial applications, triaxial testing techniques have been developed. Triaxial tests indicate a series of tensile tests on prenotched specimens in which different stress triaxiality levels are obtained by varying the notch radius of the specimen (Hancock and Mackenzie, 1976; Mackenzie et al., 1977; Hancock and Brown, 1983; Thomson and Hancock, 1984; Du and Hancock, 1991; Alves and Jones, 1999; Bonora et al., 2005, 2006; Børvik et al., 2005). It is experimentally reported that material ductility is progressively reduced with increasing the triaxiality of the stress state. Bru¨nig et al. (2008) conducted a series of experiments on smooth and prenotched tension specimen covering a wide range of stress triaxiality. In the case of the almost zero stress triaxiality, a circular channel is designed in the middle of the shear specimen to obtain a localized shear stress with zero triaxiality (Bru¨nig et al., 2011a,b). Bonora et al. (2005, 2006) conducted a series of ductile damage experimental measurements in low alloy steels under various stress triaxiality conditions and proposed a procedure to identify the CDM model parameters from experimental data. The split Hopkinson-Bar method (Kolsky, 1949) is a well-known testing technique to achieve very high strain rates and it has been extensively utilized to investigate the energy absorption of different material systems under high density dynamic loadings. Lindholm and Yeakley (1968) proposed hat-shaped specimen that can be utilized to measure tension or compression responses, and in compression, a gage for measuring radial strain was utilized to obtain the Poisson’s ratio. Numerical experiments were also conducted by Børvik et al. (2002, 2005, 2011), Bru¨nig et al. (2008, 2011a,b), and Bru¨nig and Gerke (2011) for notched and unnotched samples.

Dynamic fracture mechanics in rocks with application to drilling and perforation

235

This chapter aims at providing a sophisticated simulation approach for studying the effect of dynamic energy density on fracture responses in rocks. To accomplish this task, pressure and temperature sensitive elasto-plastic deformation of rocks is modeled and the damage mechanics of rocks are incorporated to study the effect of propagated tensile and compressive stress waves. The developed method can constitute a robust modeling tool for better understanding the role of loading rate and energy content of applied loading on the fracture of rocks. The proposed CDM model can be readily implemented into a commercial finite-element analysis (FEA) code that can be deployed into the design stages in different applications. The chapter is designed as follows: in the next section, the elasto-plastic model is outlined; in Section 11.3 dynamic CDM model is formulated. Section 11.4 describes the computational implementation and Section 11.5 presents the numerical simulation results and compares them with the available experimental measurements.

11.2

Kinematics of elastic and inelastic deformations in porous rocks

The governing field equations of a nonlinear continuum for a dynamic problem are: (1) local (or Cauchy) equations of motion, (2) equation of continuity, and (3) moment of momentum relationship (Fung and Tong, 2001). Consider a multiply connected region in which Γu and Γσ are, respectively, the displacement and traction defined boundary conditions with Γu - Γσ 5 [ and the rest of the boundary, including the crack and void boundaries, is assumed to be traction free. The integral form of the equation of motion for the multiply connected body can be written as Shojaei et al. (2013a) ð δ

 ð   ð ð @2 u i WdV 5 Pi 2 ρ 2 δui dV 1 Ti δui dS 1 Ti Δi dS @t V V Γσ ΓD

(11.1)

where W is the strain energy, σij is Cauchy stress tensor, Pi is the body force per unit volume, ρ (kg/m3) is the material density, and ui is the displacement vector. The vector Ti 5 σij nj is the traction vector on Γσ and nj is the unit vector normal to the surface. The first right-hand side (RHS) term in Eq. (11.1) accounts for the inertia effects in dynamic problems and the second RHS term shows the effect of the applied tractions on Γσ . The resulting new microsurfaces, due to the formation of microcracks and/or microvoids, produce new traction free boundaries, that is, ΓD , in which a discontinuity jump in the displacement field, that is, Δi , is generated along these new boundaries. The third RHS term concerns the traction changes due to nucleation of these new microsurfaces in the body (Shojaei et al., 2013a). The mechanical responses of porous rocks are complicated by interaction between solid skeleton and fluid-filled porous spaces. A representative volume element (RVE) for the porous rock is shown in Fig. 11.1A which is subjected to hydrostatic pressure Σ 5 1=3σkk , where σij denotes the Cauchy stress, and effective

236

Porous Rock Failure Mechanics

Figure 11.1 RVE (A) an undeformed RVE under compressive and shear stress, and (B) elastically unloaded RVE in which plasticity and damage mechanisms have changed the RVE hydro-mechanical properties.

shear stress jτ j (to be defined later). Fig. 11.1B schematically shows the state of a plastically deformed and damaged RVE in which microvoids and microcracks have been produced during the course of deformation. The solid skeleton undergoes anisotropic elastic and isochoric plastic deformation, and microcracking and microvoiding damage mechanisms are active within the skeleton. Excessive plastic deformation or damage may cause part of the solid skeleton to fail and produce more porosity for the RVE. As depicted in Fig. 11.1B the plastically deformed and damaged RVE has totally different hydromechanical characteristics compared with undeformed RVE. It is also worth noting that RVE may undergo dilatational deformation due to the presence of porosity. Thus, despite the isochoric plasticity assumption, the overall compressibility of the RVE is resulted from compaction or dilatation of preexisting or newly developed porosities. In order to prescribe constitutive relations for elastic, plastic, and damage deformation mechanisms, the total strain rate tensor, E_ ij , is decomposed into the elastic, E_ eij , plastic, E_ pij , and damaged, E_ D ij , strain rate components, E_ ij 5 E_ eij 1 E_ pij 1 E_ D ij

(11.2)

The elastic strain rate is defined based on Hook’s law as follows, E_ eij 5 L21 _ kl ijkl σ

(11.3)

where Lijkl is the fourth-order damaged stiffness tensor, “dot” indicates the rate. The constitutive relations for these strain components are formulated in the subsequent subsections. If the developed constitutive relations are to be applicable to a wide range of dynamic energy densities, for example, associated with different impact velocities, it is convenient to further decompose the elastic, E_ eij , plastic, E_ pij , _ _ kk ) and deviatoric (γ_ ij ) and the damage, E_ D ij , strain rates into their dilatational (M 5 E components,

Dynamic fracture mechanics in rocks with application to drilling and perforation

E_ ij 5 γ_ ij 1

1 _ M δij 3

237

(11.4)

where “#” may be replaced by e, p or D. Eq. (11.4) allows the deformation and the damage mechanisms, induced by the hydrostatic and deviatoric part of the applied stresses, to be formulated separately (Shojaei et al., 2013a).

11.3

Equation of state

In this work it is assumed that the induced impact and blast shock waves within the material result in a high level of the hydrostatic pressure, for example, higher than the strength of the materials by several orders of magnitude. In such circumstances an appropriate EOS is required to prescribe the relation between the pressure, that is, Σ, and the dilatational strain, that is, M. The well-known Mie-Gru¨neisen EOS is utilized in this work because its material parameters are available in the literature for a wide range of materials, see for example Steinberg (1996). For positive nominal volumetric compressive strain, that is, M 5 1 2 ðρ=ρ0 Þ . 0, where ρ0 and ρ are the initial and current densities, respectively; the linear Mie-Gru¨neisen EOS relationship reads Σ5

  ρ0 c20 M 1 Γ 1 2 M 1 Γ0 Wm 0 2 ð12sM Þ2

(11.5)

where ρ0 c20 is equivalent to the elastic bulk modulus, that is, KðMPaÞ at small nominal strains, c0 ðm=sÞ is the bulk speed of sound in the medium, Γ0 is a material constant, and Wm ðJÞ is the internal energy. The parameter s in Eq. (11.5) is the linear Hugoniot slope coefficient which defines the relationship between the shock velocity, that is, vs , and the particle velocity, that is, vp , as follows: vs 5 c0 1 sup . EOS for the reflected tensile wave, that is, M , 0, is given by Shojaei et al. (2013a), Σ 5 KM 1 Γ0 Wm

(11.6)

It is worthwhile to indicate that the EOS is only applicable in the case of high energy dynamic problems, such as high speed impacts or perforation problems, whereas in the case of low energy densities, such as low velocity impact, the hydrostatic pressure can still be defined based upon the Cauchy stresses, Σ 5 ð1=3Þσkk . From vs 5 c0 1 sup it is seen that during the elastic dilatational shock propagation, the shock speed is always greater than the bulk speed, that is, vs . c0 . Upon reflection, the elastic dilatation wave will propagate at the bulk sound speed. In this case the equation of state should be modified by setting: s 5 0. This concept is shown in Fig. 11.2 where a projectile-target impact problem is depicted schematically. The projectile approaches the target with a velocity of vp . Thereafter, impact Fig. 11.2B shows the resultant compressive propagating wave with a velocity of vs

238

Porous Rock Failure Mechanics

Figure 11.2 (A) Schematic of a projectiletarget configuration, (B) propagation of a compressive wave along the y-direction at x 5 0, and (C) reflection of tensile wave from the free surface. The pink arrows in (B) and (C) indicate the compressive and tensile nature of the propagated wave (Shojaei et al., 2013a).

within the target thickness, and Fig. 11.2C represents the reflected tensile wave from the free surface. If the maximum peak tensile stress in the reflected wave is greater than failure stress of the material, the spallation may occur at a distance δ from the free surface, see Fig. 11.2C, see Shojaei et al. (2013a) for more details.

11.4

Low-to-high strain rate constitutive modeling

Rocks, similar to frictional materials, may exhibit pressure-dependent yield behavior in which rock sample becomes stronger as the pressure increases. Also, the strain rate may have a profound effect on strain hardening and yield strength of the rock samples. In the case of dynamic problems with high energy densities, the internal heating due to dissipative mechanisms may lead to excessive elevated temperatures. The hydro-mechanical properties of rock are temperature dependent and may be affected by the generated dissipative heat. Thus, a successful dynamic constitutive law for porous rocks should take into account the effects of (1) pressure dependency, (2) strain (loading) rate, and (3) temperature. One now focuses on the rate and temperature effects first. The temperature dependent viscoplastic responses of a rock system are defined via an empirical constitutive law, namely JohnsonCook (JC). Three effects are superimposed in JC model which are: strain hardening effect, strain rate sensitivity, and the temperature change effects. The von Mises flow stress (Equivalent stress), that is, jτ j, is then defined by Johnson and Cook (1985),   p   p n 

γ_ jτ j 5 A 1 Bγ  3 1 1 Cln 3 1 2 T m γ_ 0 

(11.7)

  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where γ p  5 2=3γ pij γ pij is the equivalent plastic strain, γ_ p  5 2=3γ_ pij γ_ pij is the equivalent plastic strain rate, γ_ 0 (s21) is a reference strain rate, T  is the

Dynamic fracture mechanics in rocks with application to drilling and perforation

239

homologous temperature and AðMPaÞ; BðMPaÞ; C; n and m are five material parameters which are available for a wide set of metals in the literature (Johnson and Cook, 1985). The homologous temperature T  is given by 8 0 > > < T 2 Ttran T 5 T 2 T melt tran > > : 1

for T , Ttran for Ttran , T , Tmelt

(11.8)

for T . Tmelt

where T (K) is the current temperature, Ttran ðKÞ is the transition temperature which shows the flow stress is temperature independent for any T , Ttran and Tmelt indicates the melting temperature. Although JC model can successfully simulate the rate and temperature effects, however, it does not have any anisotropy features. Directional-dependent plastic deformations in rocks can be linked to the inherent anisotropy in the microstructure of porous rocks and anisotropic damage mechanisms. Then JC model needs to be extended to include the anisotropy effects. In the simplest form, the JC model can be introduced in an FEA model with predefined material orientations and let the FEA software account for anisotropic plastic stress fields. The von Mises yield criterion reads jτ j 2 jτ juP # 0

(11.9)

where jτ juP is the pressure sensitive strength of the rock skeleton, and it is described in Section 11.5. It is worthwhile noting that the stiffness of the rock sample is gradually degraded during the course of deformation through a CDM model, and consequently elasticplastic deformations and damage mechanisms are implicitly coupled.

11.5

Mean stress effect on the strength of rock and mechanical properties

The effect of pressure on the constitutive behavior of porous rocks is now considered. The mean stress has a significant effect on the rock constitutive behavior; consequently, the state of the applied stress triaxility significantly affects the deformation mechanisms in porous rocks. The rocks exhibit lower strength in the case of applied tensile pressures; whereas they have superior strength when they are subjected to the compressive pressure fields. On the other hand, their mechanical properties, such as shear and bulk moduli, are affected by the state of the applied pressure. In this chapter, the empirical relations are developed to account for the pressure effect on the (1) strength and (2) elastic properties of rocks in the following. The sensitivity of the rock strength jτ juP to the confining pressure field can be defined on the basis of the Homquist and Johnson model (Holmquist and Johnson,

240

Porous Rock Failure Mechanics

2005, 2008). An intact rock shows zero strength to tensile pressures greater than the rock dilatational tensile strength T (MPa), and the strength of the rock is assumed to be linearly increased from zero to jτ ju0 when the mean stress (pressure) is changed from tensile 2 T up to the compressive jΣj0 pressure. In the case of any compressive pressure greater than jΣj0 the strength is given by (Shojaei et al., 2014a):

jτ juP 5 jτ ju0 1 jτ jumax 2 jτ ju0 ½1 2 expð 2 aðΣ 2 jΣj0 ÞÞ; for Σ . jΣj0

(11.10)

where jτ jumax is the maximum achievable rock strength at high compressive pressures and the constant a is given by (Shojaei et al., 2014a): a5

jτ jumax

jτ ju0

2 jτ ju0 ðjΣj0 1 T Þ

(11.11)

The material parameters in Eq. (11.10) are readily calibrated with respect to the triaxial test data, and Eqs. (11.10) and (11.11) provide a simple and efficient representation of the pressure sensitivity for the rock strength. It is worthwhile noting that both jτ ju0 and jτ jumax parameters obey Eq. (11.7), and they are functions of stress triaxiality. The coupling between the mechanical properties of the rock and the mean stress effect is formulated by assuming that the undamaged shear μ and bulk K moduli of the rock obey the following correlations: μ 5 fp μ0 and K 5 fp K0

(11.12)

where E0 and K0 are the reference tensile and bulk elastic moduli at zero mean stress, respectively, and fp is a function to capture the effect of the pressure variations on the elastic material properties and it is represented by a linear curve fitting to the experimental data (Shojaei et al., 2014a):

E1 =E0 2 1 fp 5 1 1 Σ Σ1

(11.13)

where E1 is the tensile modulus of the rock at applied hydrostatic pressure Σ1 .

11.6

Dynamic fracture prediction techniques in continuum mechanics context

All the theoretical attempts for modeling triaxiality effects on failure of materials to date can be categorized into the following main approaches: (1) abrupt failure criteria, (2) porous plasticity mechanics with void growth models, (3) phase field modeling, and (4) CDM. The topic of dynamic crack growth has been investigated by many

Dynamic fracture mechanics in rocks with application to drilling and perforation

241

researchers including (Glennie, 1971a,b; Freund and Hutchinson, 1985; Tvergaard and Hutchinson, 1996; Landis et al., 2000; Chen and Ghosh, 2012). In general, higher crack velocity results in higher strain rate hardening effects within the fracture process zone, leading to increased toughness of the interface (Wei and Hutchinson, 1997). In this section, a CDM model is developed that takes into account the effect of dynamic energy density content of a loading scenario on the rock failure. From a multiscale analysis point of view, one may consider microscale damage mechanism dominates the failure mechanisms in porous drained rocks. In the context of CDM, the density of the microflaws are represented with damage parameters as discussed in detail by Kachanov (1958), and latter generalized to the Continuum Damage Healing Mechanics by Voyiadjis et al. (2012). Following Sayers and Kachanov (1991), Lubarda and Krajcinovic (1993), the state of anisotropic damage is introduced by a second rank damage tensor, dij . To simplify the formulation it is assumed that the anisotropic damages are mapped onto the three principal directions. The damage tensor is expressed as 2

D1 dij 5 4 0 0

0 D2 0

3 0 0 5 D3

(11.14)

where Di is the damage parameter in principal directions, to be formulated in the following. The damage parameters are correlated to the constitutive behavior of rocks through CDM. There are two cardinal damage mechanisms at microscale level within the skeleton of the porous rocks, namely microcracking and microvoiding. The microcracking damage mechanism is associated with the deviatoric part of the applied stresses in which microcracks are formed in maximum shear planes within the rock skeleton. The coalescence of these microcracks may result in failure of skeleton. Microvoiding or dilatational damage mechanism is formed in the presence of the excessive hydrostatic pressures, that is, Σ. In the case of undrained conditions, 0 the applied pore pressure p may provide enough effective tensile pressure Σ 5 Σ 2 p that can reach to a greater level than the tensile pressure strength T of the rock and volumetric damages are formed (Shojaei et al., 2014a). Rock samples may experience other damaging mechanisms such as corrosion damage in which even under low stress amplitudes, the corrosive mechanisms deteriorate the mechanical properties of the rocks. In this chapter only microcracking and microvoiding damage mechanisms are considered. The damage tensor dij is then redefined as follows, 2

Ds1

0

6 Dsij 5 4 0 0 2 d D1 6 Ddij 5 4 0

0 Dd2

0

0

Ds2 0

0

3

7 0 5; Ds3 3 0 7 0 5: Dd3

(11.15)

242

Porous Rock Failure Mechanics

where Ds represents the shear damage tensor that simulates microcracking effects, and Dd is the dilatational damage tensor that controls the rate of microvoiding in 3 principal directions (resembling an ellipsoidal void growth). The equivalence of strain energy densities between the damaged and fictitious effective configurations are utilized to establish the relations between damaged and undamaged material properties (Voyiadjis and Kattan, 2006; Voyiadjis et al., 2011, 2012; Shojaei et al., 2013a). The damaged shear modulus μ and bulk modulus K are defined based upon the undamaged shear μ and bulk K moduli, μ 5 μ ð12Ds Þ2 ; and K 5 K ð12Dd Þ2

(11.16)

Thus, the damage mechanisms in rocks reduce the elastic modulus from undamaged state to near zero. This effect will reduce the stiffness of the rock, or conversely increases its compliance. The dilatational and shear damage tensors are then introduced into the FEA software through user-defined fields. Note that the undamaged bulk modulus is initially defined by K 5 ρ0 c20 for the compressive, that is, M , 0, and K 5 K for the tensile, that is, M . 0, wave propagations within the medium. Next step is to define the damage evolution laws. Rocks contain inherent microcracks based on their mineralogical compositions and history of loading that induces an inherent anisotropy in the fracture process zone. The shear and dilatational damage parameters in three principal directions are given by (Shojaei et al., 2014a) ai 2 a0 with i 5 1; 2; and 3 ac 2 ao Vi 2 V0 Ddi 5 with i 5 1; 2; and 3 V c 2 V0 Dsi 5

(11.17)



PN ^ ki is the averaged microcrack length in the ith direction where ai 5 1=N k51 a and is called “the representative microcrack,” a0 is the initial length of the microflaws, and ac is the critical crack length at which unstable fracture occurs. In the

PN k V^ i , initial void same manner representative microvoid volume Vi 5 1=N k51

volume, V0 and critical void volume Vc are defined. The evolution of microcracks and microvoids in each of the three principal directions can be prescribed through stress-, strain-, or energy-based constitutive relationships. In the case of stress based approach, the microcracking phenomenon, that is, a^ ki , is linked to the amount of resolved shear stresses on principal k planes, and microvoiding, that is, V^ i , is correlated to effective hydrostatic pres0 sure Σ .

Dynamic fracture mechanics in rocks with application to drilling and perforation

243

Figure 11.3 Schematic representation of the damage level variation with respect to the level of the stress triaxiality. Three loading scenarios are schematically depicted: Loading scenario 1: The applied tensile pressure is high enough to cause spall fracture, Loading scenario 2: The void nucleation and propagation is the dominant damage mechanism, and Loading scenario 3: Due to the high level of the applied deviatoric stresses, the microcrack nucleation and propagation is the major damaging mechanism.

11.7

Failure modes associated with dynamic problems

Based on the state of the stress triaxiality, that is, λ 5 Σ=jτj, five regimes for the dynamic damage mechanisms in porous rocks may be considered. Let λ1 , λ2 , and λ3 , which are material parameters, define the boundaries between these regimes that are available in the literature for different materials (Bao and Wierzbicki, 2005). As shown in Fig. 11.3, these five regions are classified as follows: G

G

G

G

Regime (I) with λ , 2 λ1 : Compaction of the microvoids and microcracks, due to the applied high compressive hydrostatic stress, results in no microflaw nucleation or propagation. Regime (II) with 2 λ1 , λ , 0: Frictional sliding may occur during the microcracking process and the Coulomb friction coefficient, μðcÞ , comes into play. The microcrack growth rate equation should take into account this frictional force (Shojaei et al., 2013a). One may assume there is no void growth in this region. Regime (III) with 0 , λ , λ2 : In this regime the microcrack surfaces lose their contact and they propagate without sliding friction effect. Due to the low level of the tensile hydrostatic pressure, it is assumed that still no void can form or propagate. Regime (IV) with λ2 , λ , λ3 : Due to the high level of the applied tensile pressure in this regime, the dominant degradation mechanism is the microvoid growth. In this region the microcracking process may be neglected.

244

G

Porous Rock Failure Mechanics

Regime (V) with λ . λ3 : The spall fracture occurs at very high tensile hydrostatic pressure fields (Shojaei and Li, 2013) and it may require a separate set of damage equation to account for the tensile spall failure mode.

11.8

Constitutive relations for microcracks and microvoids

The microcrack evolution law in this work follows a strain-based criterion. The plasticity theory is formulated based on deviatoric stresses and it captures the inelastic deformation mechanisms due to shear driven stresses (Shojaei et al., 2014a). Thus, accumulated effective plastic strain corresponds directly to the level of applied shear stresses. The constitutive relation for the microcracking reads,   XN ΔEp  k a^ i 5 a0 1 χ5 ðac 2 a0 Þ k50 Ef

(11.18)

where k indicates the load step, Ef is the failure strain in which unstable crack growth occurs, and χ5 is the proportionality factor and defined as (Shojaei et al., 2014a), 8   < χ a^ i ; 6 ac χ5 5 : χ6 ;

for a^ i , ac

(11.19)

for a^ i $ ac

parameter χ6 is a material constant to calibrate the effect of microcracks interactions   in the unstable fracture process. The equivalent plastic strain increment, ΔEp , and the final fracture strain Ef are defined as follows (Shojaei et al., 2014a),   ΔEp  5

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p p ΔE ΔE ; 3 ij ij



Ef 5 d1 1 d2 expð 2 d3 λÞ ;

(11.20)

where d1 , d2 , and d3 are material constants (available from the literature, for example, Addessio and Johnson (1990), Steinberg (1996)), and λ denotes the state of the stress triaxiality, that is, λ 5 Σ=jτj. On the other hand, it is convenient to relate the microvoiding process to the level of applied hydrostatic pressure Σ. The constitutive relation for the microvoiding process is then given by, V^ i 5 V0 1 χ5 ðVc 2 V0 Þ

XN ΔΣk k50 Σ f

(11.21)

Dynamic fracture mechanics in rocks with application to drilling and perforation

245

where ΔΣk is the incremental hydrostatic pressure applied at load step k, and Σf is the void strength. The same proportionality factor χ5 is considered for both microcracking and microvoiding herein, but in general they can be different. The final task is to define the damage initiation criterion. The microcrack initiation criterion can be introduced based upon either a stress based or a fracture mechanics based criterion. Quadratic stress based criteria are well-known, and they can effectively predict the damage initiation in an intact medium. Upon fracture commencement, the stress singularity at the tip of the cracks limits the applicability of the stress based criterion. To resolve this issue, fracture mechanics based criteria have been developed in which the rate of the dissipated energy controls the damage and fracture processes (Shojaei et al., 2013a, 2014b, 2015). It is worthwhile noting that incorporating a characteristic length scale and averaging the stress at the fracture edges can resolve the stress singularity issue for stress based criterion, see Camanho and Matthews (1999), Camanho and Davila (2002), Borg et al. (2001, 2002, 2004); however, this approach encounters with computational difficulties associated with the averaging techniques. In addition, the characteristic distance is a function of geometry and material properties, so its determination always requires extensive testing. To resolve this issue, another initiation criterion is required to identify the initiation process based upon fracture energy release rates in a cracked medium (Reeder, 1992). The crack pop-up process is more complicated under mixed-mode loading conditions, in which modes-I, -II, and -III of fracture may interact. A quadratic interaction of the pure fracture modes is utilized herein to account for the mixed-mode failures. Consequently, the stress based criterion considers an intact rock, whereas the fracture mechanics criterion assumed there is a crack in the element and it rules the damage evolution (Voyiadjis and Kattan, 2014). The damage criterion is represented through an interactive description between stress based fσD and fracture mechanics based fGD descriptions as follows: for an intact rock the damage initiation criterion FD is given by,  2 FD 5 fσD 5

 2  2  2  2 1 σσ33 1 ττ12 1 ττ13 1 ττ23 0 1 0 1 0 1 τ τ τ τ τ τ 12 13 12 23 13 23 1 @ 2 A 1 @ 2 A 1 @ 2 A # 1 τ τ τ σ11 σ

 2 σ22 σ

1

(11.22)

where σ and τ  are respectively the normal and shear strengths for the damage initiation in rocks and can be obtained through experimental testing. In the case of a partially fractured rock element, the fracture mechanics based criterion FD is utilized,  2 FD 5 fGD 5

GI GIc

0

 1

GII GIIc

2

 1

GIII GIIIc

2

0

1

GI GII A 3 GIc GIIc 1

1@

1 0 G G I III A 1 @ GII 3 GIII A # M 1@ 3 GIc GIIIc GIIc GIIIc

(11.23)

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Porous Rock Failure Mechanics

where Mð , 1Þ is a material parameter, GIc , GIIc , and GIIIc are mode-I, -II, and -III critical energy release rates, respectively; and fracture energy release rates Gi i 5 I,II, and III are computed based upon the stressdisplacement as follows: Gi 5

ð δi

τ i dδi with i 5 I; II; and III

(11.24)

0

where δi is the crack opening displacement and τ I is the normal traction to the plane of fracture, and τ II and τ III are shear tractions in plane of fracture. Eqs. (11.22) and (11.23) ensure that the stress singularity issue is omitted, see Shojaei et al. (2013b) for details.

11.9

Numerical implementation, results, and discussions

The developed constitutive relations for the plasticity and damage mechanics of rocks are calibrated with respect to the triaxial tests and then they are introduced into the FEA software ABAQUS through user defined coding. The FEA model represents a homogenized porous rock system. The computational aspects of the coupled elastoplastic-damage analysis with FEA implementation are outlined as follows: G

G

G

G

STEP 1: FEA software computes the stress and strain fields for the current time step. STEP 2: Hydrostatic and deviatoric part of the computed stresses in STEP 1 are utilized to check the yield criterion and calculate the amount of plastic deformation, if yield happened. STEP 3: If the damage criterion is satisfied, deviatoric stress and plastic strain tensors are utilized to update the microcracking and microvoiding state variables, accordingly. STEP 4: Shear and dilatational damage parameters are updated based upon computed microcrack and microvoid growths in STEP 3. The Stiffness matrix for the rock system is updated for the next time increment.

The material points that exceed critical damage threshold can be removed from the FEA model through mesh deletion techniques. To minimize the mesh sensitivity of the simulations a material length scale (here element size) can be incorporated into the CDM formulation. Also, incorporating fine meshes together with enhanced hourglass property can enhance the mesh refinement convergences. The performance of the developed framework is examined in this section in simulating bit balling phenomena. The experimental data for brown colored sandstone, reported by Khazraei (1995), are utilized to verify the robustness of the plasticity and damage models, and then they are utilized to simulate the bit balling. At the first stage, the pressure sensitive strength of the rock is examined. Fig. 11.4 depicts the pressure sensitivity effect on the rock strength in which the strength of rock is studied under various applied pressures (compressive pressure is denoted by positive Σ, herein). As confirmed by the experimental results, rocks show superior strength under compressive stress fields; whereas they show little

Dynamic fracture mechanics in rocks with application to drilling and perforation

247

Figure 11.4 Mean stress effect on the strength of the rock, experiments are after Khazraei (1995), Shojaei et al. (2014a).

Table

11.1

Parameters for the pressure sensitive strength,

Eq. (11.9) jτ jumax (MPa)

jτ ju0 (MPa)

jΣj0 (MPa)

T (MPa)

180

70

5

210

resistance when they are subjected to the tensile or shear failure, like the hydraulic fracturing case. Table 11.1 summarizes the material parameters that are used in computations of pressure sensitive strength. Variation of the tensile modulus of the rock sample with the pressure is investigated in Fig. 11.5. The proposed linear curve fit to the experimental data performs accurately in capturing the mid-range data. Table 11.2 shows the material parameters that are used to obtain the data in Fig. 11.5. The elasto-plastic deformation of sandstone has been experimentally investigated by Khazraei (1995) where triaxial tests were carried out to study the deformation mechanisms of rocks under various confining pressure effects. The modified Johnson Cook Plasticity (JCP) and Johnson Cook Damage (JCD) models are calibrated with respect to the triaxial test data and Table 11.3 indicates the material parameters for elasto-plastic and damage simulations. Fig. 11.6 depicts the performance of modified JCP model with respect to experiments. Damaged elastic modulus of the rock is depicted in Fig. 11.7 where both experimental and simulation data are illustrated. The interaction between a conventional polycrystalline diamond compact (PDC) drill bit and rock sample is considered next, to illustrate the performance of the dynamic CDM model in capturing dynamic failure mechanisms in porous rocks.

248

Porous Rock Failure Mechanics

Figure 11.5 Confining stress effect on tensile modulus of the rock, experiments are after Khazraei (1995), Shojaei et al. (2014a).

Parameters for the pressure sensitive tensile modulus, Eq. (11.11) Table 11.2

E1 (GPa)

Σ1 (MPa)

E0 (GPa)

v

2.65

40

2.5

0.3

Table 11.3

Parameters for the pressure sensitive JCP and JCD

models B (MPa)

n

m

d1 (%)

d2 (%)

d3

10

0.8

1

10

10

0.05

Roller cone bits and their counterpart, PDC bits are two major drilling bits used in drilling deep hydrocarbon wells. Drag bits use a scraping, or shearing, action to remove rock. The primary drag bit in use today is PDC bits. PDC bits have cutters that are composed of a manufactured diamond table joined to a metal body, as shown in Fig. 11.8A. The cutters are the part on the bit designed to shear the rock. There are numerous attempts in the literature to develop analytical solution for dissipated mechanical energy estimations. For example, slip plane models, for example, single or multiple shear slip plane models demonstrated in Fig. 11.8B that cannot be accurate for deep drilling processes because the shearing mainly occurs in ductile materials as brittle materials fail due to compression ahead of the cutter (Astakhof, 1999; Rafatian et al., 2010). Also, existence of a single shear plane is

Dynamic fracture mechanics in rocks with application to drilling and perforation

249

Figure 11.6 Pressure sensitive stressstrain responses of porous rocks, experiments are after Khazraei (1995) and our presented numerical model which follows experimental data closely (Shojaei et al., 2014a).

Figure 11.7 Variation of elastic tensile modulus of a rock due to accumulative damages, experiments are after Khazraei (1995), Shojaei et al. (2014a).

impossible because this configuration requires unrealistically high stress gradient in the slip plane due to instantaneous formation of the chip (Ledgerwood, 2007). On the other hand, due to the large fracture process zone and complicated nature of failure ahead of the cutter, linear fracture mechanics, or more sophisticated models like cohesive cracks models, are not readily applicable for simulating the rock failure mechanisms. In the cohesive crack models, a very tortuous crack with the adjacent microcracking and frictional slips is replaced by an ideal straight line

250

(A)

Porous Rock Failure Mechanics

(B) v

Chip formation in singleshear plane model Chip φ t1

Shear planes

Drill bit

t2 γ

Work piece

Figure 11.8 (A) Halliburton’s GeoTech Fixed Cutter PDC Bits (Halliburton.com, 2016), and (B) classical shear plane analysis techniques for bit/rock interactions studies (Rahmani, 2013).

crack, which introduces a large error in determining formed chip geometries and specific energy. Also, the pattern of laying cohesive elements will govern the size and shape of formed chips. The developed CDM, on the other hand, provides a robust numerical approach that can simulate the effect of cutting mechanics on drilling efficiency. The FEA model can be utilized to measure how much of the supplied energy is dissipated due to bit balling effects, and the rest is consumed due to rock failure mechanisms. It is apparent that higher bit balling effects mean less drilling efficiency (more energy is consumed by cuttings instead of intact rock) and vice versa. The proposed failure model is introduced into a commercial FEA package, ABAQUS (2013), through user-defined subroutines, that is, User Defined Field Variable (VUSDFLD) and User Defined material (VUMAT). The detail for computational aspects are discussed elsewhere and an interested reader may refer to Shojaei et al. (2014a) and Shojaei et al. (2010). A 2D FEA model is constructed to study the contact mechanics between a rigid PDC bit and rock sample. Normal hard and tangential frictionless contact properties are defined together with the adaptive Lagrangian Eulerian remeshing rule to enhance the progressive mesh removal algorithm. The rock is exposed to downhole pressure effects and a constant velocity (rpm) is considered for the bit. As depicted in Fig. 11.9B and C, the progressive erosion of rock’s elements is well captured by the FEA model. Several parameters that might affect the bit’s performance can be studied by these FEA models including the back rake angle, γ, relative cutting to rock depth thickness ratios and rock angle, φ (Shojaei and Dahi, 2016). As depicted in Fig. 11.10, the mesh refinement study is carried out by Shojaei and Dahi (2016). The experimental data was collected from published single cutter test under a confining pressure of 1000 psi, reported by Smith (1998). The four data point shown in Fig. 11.10 illustrate the recorded force vs depth of drilling from the onset of cutting at 0.025v, 0.05v, 0.075v, and 0.1v depths of cuts.

Dynamic fracture mechanics in rocks with application to drilling and perforation

251

Figure 11.9 Simulation of contact between a rigid PDC drill bit and rock, (A) boundary conditions, and (B) and (C) show the progressive rock’s failure ahead of the drill bit.

11.10

Conclusion

The anisotropic plasticity and damage mechanisms in rocks are studied and a new CDM model is developed for dynamic fractures of porous rocks. The effect of stress triaxility on Rock’s elasto-plastic behavior is discussed. JC’s law is incorporated into the proposed viscoplasticity to consider the rate, temperature, and triaxility effects. Elasticity and plasticity constitutive relations are interrelated to the damage mechanisms in rocks through the CDM framework. The effect of hydrostatic (confining) pressure on the plastic deformation and mechanical properties of rocks are also considered. The plasticity model correlates well with the available experimental data in the literature.

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Figure 11.10 Mesh refinement study for 1.0, 0.5, 0.25, and 0.125 mm relative sizes (Shojaei and Dahi, 2016). Experimental data is after Smith (1998).

The CDM model considers both shear driven and dilatation damages. The microcracking mechanisms, driven by the shear stresses, are utilized to formulate the shear-driven CDM model. The microvoiding mechanisms are linked to the dilatational damages. Two stress based and fracture mechanics based damage initiation criteria are presented to accurately control the rate of damage progress and resolve the stress singularity issue in FEA problems with progressive damage. The developed computational platform has been implemented in FEA software to study sundry problems. As an example, a drill bit-rock interaction problem is considered and it is shown that the developed framework is in good agreement with the reported experimental data. The developed framework can be utilized to construct a virtual lab to better understand the dynamic fracture responses of rock samples under confining pressures.

References ABAQUS, 2013. Abaqus 6.13 Documentation. Dassault Systemes Simulia Corp, Rhode Island, USA. Addessio, F.L., Johnson, J.N., 1990. A constitutive model for the dynamic response of brittle materials. J. Appl. Phys. 67 (7), 32753286. Alves, Ml, Jones, N., 1999. Influence of hydrostatic stress on failure of axisymmetric notched specimens. J. Mech. Phys. Solids. 47 (3), 643667. Astakhof, V.P., 1999. Metal Cuttings Mechanics. CRC press. Bao, Y., Wierzbicki, T., 2005. On the cut-off value of negative triaxiality for fracture. Eng. Fract. Mech. 72 (7), 10491069. Bonora, N., Gentile, D., Pirondi, A., Newaz, G., 2005. Ductile damage evolution under triaxial state of stress: theory and experiments. Int. J. Plast. 21 (5), 9811007.

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Bonora, N., Ruggiero, A., Esposito, L., Gentile, D., 2006. CDM modeling of ductile failure in ferritic steels: assessment of the geometry transferability of model parameters. Int. J. Plast. 22 (11), 20152047. Borg, R., Nilsson, L., Simonsson, K., 2001. Simulation of delamination in fiber composites with a discrete cohesive failure model. Compos. Sci. Technol. 61 (5), 667677. Borg, R., Nilsson, L., Simonsson, K., 2002. Modeling of delamination using a discretized cohesive zone and damage formulation. Compos. Sci. Technol. 62 (1011), 12991314. Borg, R., Nilsson, L., Simonsson, K., 2004. Simulation of low velocity impact on fiber laminates using a cohesive zone based delamination model. Compos. Sci. Technol. 64 (2), 279288. Børvik, T., Hopperstad, O.S., Berstad, T., Langseth, M., 2002. Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses: Part II: Numerical simulations. Int. J. Impact Eng. 27 (1), 3764. Børvik, T., Hopperstad, O.S., Dey, S., Pizzinato, E.V., Langseth, M., Albertini, C., 2005. Strength and ductility of Weldox 460 E steel at high strain rates, elevated temperatures and various stress triaxialities. Eng. Fract. Mech. 72 (7), 10711087. Børvik, T., Olovsson, L., Hanssen, A.G., Dharmasena, K.P., Hansson, H., Wadley, H.N.G., 2011. A discrete particle approach to simulate the combined effect of blast and sand impact loading of steel plates. J. Mech. Phys. Solids. 59 (5), 940958. Bru¨nig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. Int. J. Plast. 27 (10), 15981617. Bru¨nig, M., Albrecht, D., Gerke, S., 2011a. Modeling of ductile damage and fracture behavior based on different micromechanisms. Int. J. Damage Mech. 20 (4), 558577. Bru¨nig, M., Albrecht, D., Gerke, S., 2011b. Numerical analyses of stress-triaxialitydependent inelastic deformation behavior of aluminum alloys. Int. J. Damage Mech. 20 (2), 299317. Bru¨nig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M., 2008. A ductile damage criterion at various stress triaxialities. Int. J. Plast. 24 (10), 17311755. Camanho, P.P., Davila, C.G., 2002. Mixed-Mode Decohesion Finite Elements for the Simulation of Delamination in Composite Materials. NASA/TM-2002211737. NASA137. Camanho, P.P., Matthews, F.L., 1999. A progressive damage model for mechanically fastened joints in composite laminates. J. Compos. Mater. 33 (24), 22482280. Chen, Y., Ghosh, S., 2012. Micromechanical analysis of strain rate-dependent deformation and failure in composite microstructures under dynamic loading conditions. Int. J. Plast. 3233 (0), 218247. Cox, B.N., Gao, H., Gross, D., Rittel, D., 2005. Modern topics and challenges in dynamic fracture. J. Mech. Phys. Solids. 53 (3), 565596. Dai, F., Xia, K., Zheng, H., Wang, Y.X., 2011. Determination of dynamic rock Mode-I fracture parameters using cracked chevron notched semi-circular bend specimen. Eng. Fract. Mech. 78 (15), 26332644. Du, Z.Z., Hancock, J.W., 1991. The effect of non-singular stresses on crack-tip constraint. J. Mech. Phys. Solids. 39 (4), 555567. Fahrenthold, E.P., 1991. A Continuum damage model for fracture of brittle solids under dynamic loading. J. Appl. Mech. 58 (4), 904909. Freund, L.B., Hutchinson, J.W., 1985. High strain-rate crack growth in rate-dependent plastic solids. J. Mech. Phys. Solids. 33 (2), 169191.

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Rahmani, R., 2013. Analytical Modeling and Diagnosis of Penetration Rate Performance of PDC Bits. Department of Petroleum Engineering,. Louisiana State University, USA. Reeder, J.R., 1992. An Evaluation of Mixed-Mode Delamination Failure Criteria. NASA TM 104210, NASA. Sayers, C.M., Kachanov, M., 1991. A simple technique for finding effective elastic constants of cracked solids for arbitrary crack orientation statistics. Int. J. Solids Struct. 27 (6), 671680. Shojaei, A., Eslami, M., Mahbadi, H., 2010. Cyclic loading of beams based on the Chaboche model. Int. J. Mech. Mater. Design. 6 (3), 217228. Shojaei, A. and A.D. Taleghani, 2016. A Continuum Damage Model to Predict Bit Performance Based on Single Cutter Experiments. 13th ISRM International Congress of Rock Mechanics, Montreal, Canada, Published by International Society for Rock Mechanics. Shojaei, A., Li, G., 2013. Viscoplasticity analysis of semicrystalline polymers: A multiscale approach within micromechanics framework. Int. J. Plast. 42, 3149. Shojaei, A., Voyiadjis, G.Z., Tan, P.J., 2013a. Viscoplastic constitutive theory for brittle to ductile damage in polycrystalline materials under dynamic loading. Int. J. Plast. 48, 125151. Shojaei, A., G. Li, P.J. Tan and J. Fish, 2013b. Dynamic energy density effect on the delamination damage in laminated fiber reinforced polymers Under review in the J. Appl. Mech. Shojaei, A., Dahi Taleghani, A., Li, G., 2014a. A continuum damage failure model for hydraulic fracturing of porous rocks. Int. J. Plast. 59 (0), 199212. Shojaei, A., Li, G., Fish, J., Tan, P.J., 2014b. Multi-scale constitutive modeling of ceramic matrix composites by continuum damage mechanics. Int. J. Solids Struct. 51 (2324), 40684081. Shojaei, A., Li, G., Tan, P.J., Fish, J., 2015. Dynamic delamination in laminated fiber reinforced composites: A continuum damage mechanics approach. Int. J. Solids Struct. 71, 262276. Slepyan, L.I., 2010. Dynamic crack growth under Rayleigh wave. J. Mech. Phys. Solids. 58 (5), 636655. Smith, J.R., 1998. Diagnosis of poor PDC bit performance in deep shales. Louisiana State University, Baton Rouge, USA, PhD Dissertation. Steinberg, D.J., 1996. Equation of State and Strength Properties of selected Materials. Livermore, CA, Tech.Rep.UCRL-MA-106439, Lawrance Livermore National Labratory, Livermore, CA. Thomson, R.D., Hancock, J.W., 1984. Ductile failure by void nucleation, growth and coalescence. Int. J. Fract. 26 (2), 99112. Tvergaard, V., Hutchinson, J.W., 1996. Effect of strain-dependent cohesive zone model on predictions of crack growth resistance. Int. J. Solids Struct. 33 (2022), 32973308. Voyiadjis, G.Z., Kattan, P.I., 2014. How a singularity forms in continuum damage mechanics. Mech. Res. Commun. 55, 8688. Voyiadjis, G.Z., Shojaei, A., Li, G., 2011. A thermodynamic consistent damage and healing model for self healing materials. Int. J. Plast. 27 (7), 10251044. Voyiadjis, G.Z., Shojaei, A., Li, G., Kattan, P.I., 2012. A theory of anisotropic healing and damage mechanics of materials. Proc. R. Soc. A: Math. Phys. Eng. Sci. 468 (2137), 163183. Voyiadjis, Z., Kattan, P.I., 2006. Advances in Damage Mechanics. Elsevier, London.

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Wang, S.Y., Sun, L., Yang, C., Yang, S.Q., Tang, C.A., 2013. Numerical study on static and dynamic fracture evolution around rock cavities. J. Rock Mech. Geotech. Eng. 5 (4), 262276. Wei, Y., Hutchinson, J.W., 1997. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. J. Mech. Phys. Solids. 45 (8), 12531273. Wu, Z., Liang, X., Liu, Q., 2015. Numerical investigation of rock heterogeneity effect on rock dynamic strength and failure process using cohesive fracture model. Eng. Geol. 197, 198210. Xu, X.P., Needleman, A., 1994. Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids. 42 (9), 13971434. Yang, R., Bawden, W.F., Katsabanis, P.D., 1996. A new constitutive model for blast damage. Int. J. Rock Mech. Mining Sci. Geomech. Abstracts. 33 (3), 245254. Yang, R.-s, Yue, Z.-w, Sun, Z.-h, Xiao, T.-s, Guo, D.-m, 2009. Dynamic fracture behavior of rock under impact load using the caustics method. Mining Sci. Technol. (China). 19 (1), 7983. Yazdchi, M., Valliappan, S., Zhang, W., 1996. A continuum model for dynamic damage evolution of anisotropic brittle materials. Int. J. Numer. Methods Eng. 39 (9), 15551583. Zhang, Z., Yao, Y., Mao, X., 2015. Modeling wave propagation induced fracture in rock with correlated lattice bond cell. Int. J. Rock Mech. Mining Sci. 78, 262270.

Stability, accuracy, and efficiency of numerical methods for coupled fluid flow in porous rocks

12

Richard Giot and Albert Giraud University of Lorraine/CNRS/CREGU, Vandœuvre-le`s-Nancy, France

12.1

Introduction/Framework

Coupled fluid flow is a paramount physical process in the response of porous rocks in the framework of numerous engineering applications such as underground nuclear or hazardous waste storage, gas (such as CO2) storage, hydraulic fracturing (hydroshearing and hydrofracking), georesources completion (geothermal energy, oil, and gas resources), or even building and maintenance of construction sites (tunnels, dams, and mines). In addition, in all these applications, the rock mass is generally fractured. The physical processes are fully coupled and highly nonlinear, so that analytical solutions are hardly ever available or applicable. The understanding and prediction of the behavior of the rock mass and possibly the underground structures then requires numerical modeling. In the present chapter, thus, we focus on the numerical methods for coupled fluid flow in porous rocks. We propose an overview of the most widespread and recent methods, with or without accounting for fractures. Here, we will not present in details numerical methods such as finite element method (FEM), discrete element method (DEM), boundary element method (BEM), extended finite element method (XFEM), and others, which are methods classically used in solid mechanics and already detailed in the specific literature. We rather make a literature overview of methods used in the framework of coupled fluid flow in porous rocks. We focus on the development required to adapt them to this framework. We also highlight the strengths and limitations of the methods and focus on some particular points concerning the stability, accuracy, and efficiency of the numerical methods presented.

Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00012-9 © 2017 Elsevier Ltd. All rights reserved.

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12.2

Porous Rock Failure Mechanics

Numerical modeling of continuous media applied to rocks

12.2.1 Review of the numerical methods for coupled fluid flow modeling in continuous porous rocks Upon numerical methods developed for numerical modeling of coupled fluid flow in porous media, the most popular and widespread are FEM, finite volume method (FVM), and finite difference method (FDM). These are classical approaches to obtain numerical approximation of partial differential equations (PDEs) that describe the physical processes governing the coupled fluid flow in porous rocks. Some of these methods have been combined, which appears to be a sequential resolution of the set of PDEs, for example, FVM for the resolution of the fluid flow problem and FEM for the mechanical problem. Here, we do not want to present in details these different numerical methods, which were largely discussed in the literature. Instead, we propose an overview on the application of these methods to coupled fluid flow in porous rocks and present some advantages, drawbacks, and adaptation of the methods in this framework. Moreover, we chose to confine ourselves to the numerical approaches that explicitly account for mechanical and hydraulic behaviors and their couplings. Due to the presence of gas and hydromechanical couplings in rocks, the coupled diffusion process in porous rocks in both fully and partially saturated states is highly nonlinear (Olivella et al., 1994). When considering the general case of partially saturated porous media, these equations are highly nonlinear due to the presence of gas and as an consequence numerical methods, essentially finite element or FVMs, are required to solve the direct problem (Thomas et al., 1994; Gawin et al., 1995; Eymard et al., 2000; Chavant et al., 2002; Bianco et al., 2003; Schrefler, 2004; Benard et al., 2006; Gowing et al., 2006; Morency et al., 2007; Dal Pizzol and Maliska, 2012). FEM is one of the most widespread numerical methods for the numerical modeling of Hydro-Mechanical (HM) couplings in porous rocks, accounting for fluid flow and rock deformation. It has been considered by several authors in the framework of nuclear waste storage accounting for different physical processes such as elastoplasticity, damage, viscosity, two-phase flow, and thermal loading (Jia et al., 2009; Charlier et al., 2013; Levasseur et al., 2013; Salehnia et al., 2015; Pardoen et al., 2015; Gerard et al., 2014; Giraud et al., 2009; Giot et al., 2011; Giot et al., 2012; Guillon et al., 2012). FEM was also applied in the context of CO2 storage (Dieudonne´ et al., 2015; Saliya et al., 2015). Special HM coupled finite elements were developed, for example, two-dimensional (2D) second gradient finite elements to deal with localized shear zones in rocks (Collin et al., 2006) or coupled interface finite elements for contact problem (Cerfontaine et al., 2015). A special attention was also given to poroelastic anisotropy, which is of paramount importance when dealing with rocks. Cui et al. (1996) considered an anisotropic poroelastic model for finite element analysis of geomechanical problems. Noiret et al. (2011) applied a transverse isotropic poroelastic constitutive law implemented in a finite element code for the interpretation of œdometer tests on a claystone, whereas Giot et al. (2012, 2014) used a similar approach for the interpretation of axial and radial pulse tests for intrinsic permeability identification of a claystone.

Stability, accuracy, and efficiency of numerical methods for coupled fluid flow in porous rocks

259

Finite difference method was also considered for several application fields for porous rocks. Amongst many others, let us cite some recent works aiming at modeling coupled fluid flow in porous rock with FDM. Blanco-Martin et al. (2015) considered FDM in the framework of the long-term THM modeling of nuclear waste repository in salt. They used a sequential coupling between FLAC3D (Itasca), which is based on FDM scheme and solves the geomechanical problem, and TOUGH2, which solves the multiphase fluid and heat flow problem. Bian et al. (2012) also employed the FDM to model the THM response of a claystone to a thermal loading, considering a full coupling rather a staggered approach. In this work, the rock is supposed to be fully saturated. Hu et al. (2013) use an Integral Finite Difference Method for geothermal application. Other methods are proposed by several authors, such as control-volume finite element (Mello et al., 2009) which can be seen as an hybrid method based on FEM and FVM, or a coupling of lattice Boltzmann with distinct element method (Boutt et al., 2007, 2011). This last method is based on a micromechanical approach for the consideration of solid
fluid couplings. The fluid flow is accounted for through a lattice Boltzmann method, while the mechanical problem is solved with a DEM. Cavalcanti and Telles (2003) used the BEM to solve 2D problems concerning fully saturated porous media. FVM is classically used for fluid flow problem but coupled to a FEM for the modeling of rock deformation. Authors such as Li et al. (2016) propose numerical methods to improve the data transfer between both schemes.

12.2.2 Coupled fluid flow in rocks with finite element modeling in fully saturated conditions We recall here, the main equations of a fully coupled hydromechanical model for the fully saturated domain. For the simplicity of the presentation, we do not deal here with neither the thermal nor the chemical processes. We will nevertheless account for these physical phenomena in the discussions concerning the numerical implementation of the models. The HM model presented herein is based on Biot theory for saturated porous media (Coussy, 2004, see also Wang, 2001; Cheng, 2016). Let us consider a porous medium consisting of a deformable matrix fully saturated by a compressible fluid (index lq). The nonlinear poroelastic constitutive equations for the saturated case can be written incrementally: dσm 5 Ko dεv 2 bdplq dσm 5 Kun dεv 2 bM

dmlq ρlq

dsij 5 2Gdeij
 dmlq dplq 5 M 2bdεv 1 ρlq

(12.1a) (12.1b) (12.1c) (12.2)

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In these equations, σm, s, εv, e, mlq, ρlq, and plq respectively stand for mean stress, deviatoric stress tensor, volumetric strain, deviatoric strain tensor, fluid mass supply, liquid density, and liquid pressure. Ko and Kun are the compressibility moduli in drained (dplq 5 0) and undrained (dmlq 5 0) conditions. b and M are Biot coefficient and modulus of the fully saturated porous medium, respectively. From these equations, it is possible to infer the liquid mass supply as a function of liquid pressure and volumetric strain: dmlq 5 bdεv 1 Ndplq ρlq

(12.3)

with N 5 1/M. The equation ruling the deviatoric behavior is decoupled from the equation ruling the volumetric behavior. To these poroelastic constitutive equations, it is required to add conduction laws. In the following, as already mentioned, we focus on the isothermal problem, so that we neglect here the heat conduction law, which can nevertheless be taken into account in the model with no more difficulty. Considering generalized Darcy law and neglecting the effects of gravity, the liquid flow is given by M lq 5 2 λlq rplq ρlq

(12.4)

In this expression, λlq stands for Darcy conductivity for the liquid, which can be linked, for the fully saturated porous medium, to the intrinsic permeability: λlq 5

kint μlq

(12.5)

where μlq stands for dynamic viscosity of the liquid. To complete the set of PDEs, it is necessary to account for mechanical equilibrium and liquid mass conservation equations: r  σ 1 ρFm 5 0

(12.6)

@mlq 5 2 r  M lq @t

(12.7)

Initial and boundary conditions should obviously be added to these field equations.

12.2.3 Coupled fluid flow in rocks with finite element modeling in partially saturated conditions Let us consider a porous medium consisting of a deformable matrix, partially saturated by a compressible liquid (index l) in equilibrium with its vapor (index v), the latter forming an ideal mixture (index g) with another gas (dry air, index a).

Stability, accuracy, and efficiency of numerical methods for coupled fluid flow in porous rocks

ρa 5

M a pa ; RT

ρv 5

M v pv ; RT

pg 5 pa 1 pv

261

(12.8)

Ma, Mv, and R respectively stand for dry air and vapor molar mass and ideal gas constant. Vapor pressure can be removed from the set of equations through liquid water
vapor thermodynamical equilibrium equation:    pv Mv  0 pg 2 pc 2 pl ; hr 5 0 5 exp pv ρl RT

pc 5 pg 2 pl

(12.9)

In these equations, “0” exponent refers to saturated state, hr is the relative humidity, and p0v is the saturation vapor pressure. Amongst the three pressure variables (pl liquid pressure, pg, gas pressure, and pc capillary pressure), capillary and gas pressures can be chosen as independent state variables (or primary variables). Nonlinear isotropic isothermal poroelastic constitutive law for the partially saturated conditions (Coussy, 2004) is written incrementally. dσm 5 KO dεv 1 dσp ;

  dσp 5 2 b dpg 2 Sl dpc

(12.10a)

dsij 5 2Gdeij

(12.10b)

dσij 5 dσ0ij 1 dσp δij

(12.10c)

dml 1 dmv 5 Clε dεv 1 Clc dpc 1 Clg dpg

(12.11)

dma 5 Cgε dεv 1 Cgc dpc 1 Cgg dpg

(12.12)

In these equations, σm, s, σ 0 , εv, e, mi, KO, and G respectively stand for total mean stress, deviatoric stress tensor, effective stress tensor, volumetric strain, deviatoric strain tensor, constituents fluid mass supply, drained compressibility coefficient, and shear modulus. σp is the part of the mechanical stress due to saturating pores fluids, and Sl is the liquid saturation. Coupling parameters Cij are functions of saturation and its derivate with respect to capillary pressure, as well as partial pressures. Generalized Darcy law for liquid and gas advection is written as follows:   wi 5 λi 2rpi 1 ρi Fm ρi λi 5

kKirel ½Sl  μi

(12.13)

(12.14)

where k is the intrinsic permeability, and λi, wi, Kirel and μi respectively stand for conductivity, flow, relative permeability, and dynamic viscosity for fluid I

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(either l or g). Vapor diffusion in gas is accounted for through Fick law, with Fg standing for Fick coefficient for gas mixture: wv wa 2 5 2 F g rCv ; ρv ρa

Cv 5

pv pg

(12.15a)

A Fick law is also taken into account for diffusion of water and air dissolved in the liquid, with Fl Fick coefficient for liquid mixture: wda ww 2 5 2 F l rCda ; ρda ρw

Cda 5

pda pw

(12.15b)

Mass conservation equations are written for water and air: @ ½mw 1 mv  5 2 r  ½ww 1 wv  @t

(12.16a)

@ ½ma 1 mda  5 2 r  ½wa 1 wda  @t

(12.16b)

Mechanical equilibrium equation should also be accounted for r  σ 1 ρFm 5 0

(12.17)

This model thus takes into account the hydromechanical couplings. It can easily be extended to thermohydromechanical couplings. The set of PDEs must be completed with initial and boundary conditions for both mechanics and hydraulics. Due to the HM couplings, this set of PDEs does not admit, generally, an analytical solution and then must be solved numerically.

12.2.4 Numerical scheme As an illustration of a classical and efficient numerical scheme, we chose to present the numerical scheme implemented in the Finite Element code Code_Aster (Edf, see Code Aster; Granet, 2014, 2015) and used for example, and amongst many others, in Giraud et al. (2009), Giot et al. (2011), Giot et al. (2012), Guillon et al. (2012), and Saliya et al. (2015). The set of PDEs presented in the previous sections can be solved using a FEM. The first step of the numerical formulation consists in writing the equations under a variational form. The mechanical equilibrium then gives ð

ð Ω

σ  εðvÞdΩ 5

ð ρF  vdΩ 1 m

Ω

@Ω

f ext  vdΓ

’ vAUad

where Uad indicates the set of cinematically admissible displacements, and ρ is the homogenized volumic mass.

Stability, accuracy, and efficiency of numerical methods for coupled fluid flow in porous rocks

263

Mass conservation equations are also written under variational form: ð
2

Ω

 ð dmw dmv 1  Πl dΩ 1 ðww 1 wv Þ  rΠl dΩ dt dt Ω ð  ext  ww 1 wext 5  Πl dΓ ’ Πl APl ad v @Ω

ð
2

Ω

 ð dmda dma 1  Πg dΩ 1 ðwda 1 wa Þ  rΠg dΩ dt dt Ω ð  ext  wda 1 wext 5  Πg dΓ ’ Πg APg ad a @Ω

where Pl ad and Pg ad are the admissible pressure fields, and wext refers to scalar i hydraulic flows on the boundaries of the domain. A theta-scheme is applied in this approach. θ is a numerical parameter between 0 and 1. The next step is to proceed to time discretization. Since the mechanical equilibrium is not time dependent, this discretization only concerns the mass conservation equations. In the following expression the exponents “1” and “2” respectively refer to values at the end and the beginning of the time step. Moreover Δt 5 t1 2 t2 χθ 5 θχ1 1 ð1 2 θÞχ2 Time discretization for mass conservation equations is then written as ð 2





ð

 1  ww 1 w1 v  rΠl dΩ Ω Ω ð ð  2   2  52 mw 1 m2 dΩ 2 ð 1 2 θ ÞΔt ww 1 w2  Π l v v  rΠl dΩ Ω ð Ω  extθ   Π ww 1 wextθ dΓ ’ Πl APl ad 1 Δt l v 1 m1 w 1 mv

 Πl dΩ 1 θΔt

@Ω

ð 2

Ω



 1 m1 da 1 ma  Πg dΩ 1 θΔt ð

52

 Ω

ð Ω



 1 w1 da 1 wa  rΠg dΩ

 2 m2 da 1 ma  Πg dΩ 2 ð1 2 θÞΔt

1 Δt

ð @Ω



 extθ  Πg dΓ wextθ da 1 wa

ð

 Ω

 2 w2 da 1 wa  rΠg dΩ

’ Πg APg ad

The system of equations is then discretized by finite element and written as a nonlinear matrix system, which can be solved with a Newton
Raphson algorithm. The spatial discretization is based on mixed finite elements for the coupled HM

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modeling. In order to avoid spurious oscillations of the solution, particularly for small time steps, orders of approximations must be different for displacements and pressures discretizations. As a consequence, in Code_Aster (Edf), the hydromechanical elements (in fact, thermohydromechanical elements) account for a quadratic interpolation for the displacements, and a linear interpolation for the pressures (and temperature).

12.2.5 Other numerical schemes. Performances, stability, and accuracy Classically, the set of nonlinear PDEs governing the coupled flow problem is discretized with a FEM, considering appropriate discretization both in time and space, as previously illustrated. For the spatial discretization, a Galerkin finite element scheme is often used (Tong et al., 2010; Lewis and Pao, 2002). The time discretization is often based on a first-order finite difference scheme (Kolditz et al., 2012; Watanabe et al., 2010), which should be implicit to be unconditionally stable. An alternative is to consider a theta-scheme, as mentioned in Section 12.2.4, which can be seen as a ponderation between a fully explicit and implicit time discretization. According to Tong et al., depending on this ponderation, several classical schemes can be established: explicit (or backward or Euler), implicit (or forward), middifference (Cranck
Nicholson) or Galerkin. After discretization, the FEM problem is written under matrix form and results in a nonlinear asymmetric matrix system. The coefficient matrix involved in this system is often illconditioned, so that the set of equations and degrees of freedom should be properly ordered to obtain a diagonally dominant matrix, which should improve stability and rate of convergence of the FEM resolution. The FEM formulation of the set of equations ruling the coupled fluid flow requires the establishment of the weak, or variational, formulation of these equations, which can be achieved thanks to the weighted residuals method in the fully and partially saturated cases (Kolditz et al., 2012; Watanabe et al., 2010). A system of nonlinear algebraic equations must then be solved. Kolditz et al., in the framework of CO2 storage, propose a HM coupled two-phase flow model and suggest three different algorithms for the resolution of the nonlinear set of algebraic equations by FEM: Picard (fix point) linearization, mixed finite element algorithm developed by Truty and Zimmerman and a Newton algorithm. The authors underline the difficulty to set a first iterate to initiate these iterative algorithms in the context of THM problem, particularly when two-phase flow is considered. In order to ensure an equivalent numerical accuracy for the resolution of the different PDEs, the authors use linear interpolation for the hydraulic and thermic variables and quadratic interpolation for the mechanical variables, as mentioned in the case of Code_Aster in Section 12.2.4. Moreover, the authors emphasize the importance of the choice of the primary variables when dealing with two-phase flow models to ensure the stability of the numerical scheme. The temporal discretization of the coupled problem is achieved through PI (proportional and integral feedback) algorithm. This control-theoric approach allows for an optimized automatic stepsize control for the numerical integration of PDEs, which enhances robustness, stability, and accuracy of the solution, particularly when numerical stability limits the stepsize.

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Sun (2015) also highlighted the instability in the numerical response when considering interpolations of the same order for both hydraulic and mechanical primary variables. Rather than introducing two different-order interpolations for pore pressure and displacement fields, Sun develops a stabilization procedure in the framework of a THM finite element model with equal-order discretized temperature, pore pressure, and displacement fields. The suggested stabilization procedure enforces the satisfaction of a combined inf-sup condition, thus ensuring numerical stability. One of the issues when dealing with fully coupled flow in porous rock is the computational requirements of the models, both in time and memory. Some authors suggested to simplify the mathematical model previously presented, in order to reduce the computational requirements of the model. Hu et al. (2013) reduced the number of degrees of freedom of the set of equations by focusing on the mean total stress as the only geomechanical variable rather than the three components of the displacement or velocity vector in three-dimensional (3D) calculations. The rock mass is supposed to be linear elastic and its behavior is described by the generalized Hooke’s law. The space discretization is achieved through the integral finite difference method while the time integration makes use of a backward first-order finite difference scheme. Quite classically, the set of equations is written under residuals form and solved through a Newton
Raphson algorithm. The main shortcomings of this approach lies in the fact that the model can only deal with volumetric behavior and is unable to handle both shear stresses and rock failure. Another approach, to deal with the issues regarding the time and memory requirements of numerical modeling in the framework of coupled fluid flow in porous rock, is the parallelization of the Finite Element scheme (Wang et al., 2009; Watanabe et al., 2010). Wang et al. detail a method for parallelization of a finite element scheme in the framework of THM coupled process. This method uses MPI and is based on a domain decomposition, a partitioning of the global assembly and of the global linear solver. Details on the parallelization procedure, as well as discussion on accuracy and efficiency, are given in the paper of Wang et al. (2009). These discussions are out of the scope of the present chapter. Nevertheless, the authors emphasize that increasing the number of domain decomposition does not alter accuracy of the solution. The interested reader is referred to the paper for more details. For engineering applications such as CO2 storage, additional physical processes must be taken into account, including thermal and chemical phenomena, resulting in a thermohydromechanical
chemical (THMC) couplings. The set of PDEs governing these THMC couplings then forms a reactive transport problem that can be solved essentially through two different ways. Either the equations are fully coupled and the whole set of equations is solved simultaneously (Zhang et al., 2016), or the solute transport and chemical reactions equations are decoupled and the set of equations is then solved sequentially (Zhang et al., 2015). Yin et al. (2012) considered thermal and solute convection and thus proposed a sequential resolution where the coupled THMC field equations are solved with a FEM, while the chemical reactions are solved with a Newton
Raphson iterative algorithm coupled with a predictor
corrector method. For the resolution of the coupled THMC field equations, the authors highlight two kinds of numerical instabilities when solving their transient advection
diffusion problem: oscillatory instability

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in the temperature and solute concentration responses under convection
dominant regime, and instability in the first-time steps under the transient diffusion
convection regime. Depending on whether the problem evolves in a diffusion
dominated regime or in an advection dominated regime, sharp gradients appearing in the transient advection
diffusion problem will disappear or not after a limited number of time steps. In these advection
diffusion problems, the classical Galerkin FEM is not adequate to address the numerical instabilities, which the authors already dealt with in a previous paper (Yin et al., 2010). In this work, they proposed a stabilized FEM named the subgrid scale/gradient subgrid scale (SGS/GSGS) method and compared it to others formulations: the streamline upwind Petrov
Galerkin (SUPG) and the Galerkin least-squares methods, which consist in adding a stabilization term to the Galerkin formulation. The authors showed that both later approaches are efficient in steadystate advection
diffusion problems, but did not handle the instabilities in the unsteady advection
diffusion problems. The approach proposed by Yin et al. consists in first transforming the transient diffusion
convection problem to a steady diffusion
convection reaction problem by time discretization and then applying the SGS/GSGS stabilization integrals to the latter problem. The stabilization terms of the SGS/GSGS method incorporates the Peclet and Damko¨hler numbers. The numerical instabilities at small time steps are then handled. As an example, Yin et al. (2010) give concentration and pore pressure profiles obtained by both SGS/GSGS and Galerkin FEM on a problem of wellbore drilling. Yin et al. (2012) give the H 1 concentration and temperature profiles obtained by SGS/ GSGS and Galerkin FEM for a problem of injection of CO2 saturated water in a carbonate aquifer. In both examples, it is shown that the SGS/GSGS scheme allows to eliminate the numerical oscillations of the Galerkin FEM. The reader is referred to the mentioned papers of Yin et al. for more details. As already mentioned, solving the system of nonlinear equations can classically be achieved with a Newton
Raphson algorithm Callari and Abati (2009) proposed a finite element formulation with special attention and details on the consistent tangent operators appearing in Newton
Raphson algorithm. In their formulation, the tangent operators are consistent with the finite difference scheme used for time integration, rather than continuum tangents, which improves the convergence rate of Newton
Raphson algorithm.

12.3

Numerical modeling of fractured rocks

12.3.1 Literature review of different approaches for numerical modeling of fluid-driven fractures in rocks Here, we make a brief literature review of the recent works on coupled fluid flow modeling in fractured porous rock. The main idea is to present the different approaches used by several authors and to underline their main advantages and drawbacks. We will not detail the principle of each numerical method (e.g., FEM,

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DEM, discontinuous deformation analysis (DDA), partition of unity method (PUM), etc.), which can be found in the specialized literature, but rather present how these methods are adapted to the coupled fluid flow problems. The idea is to explore the different approaches and help the readers to understand the main interest of each approach to elect which one is the most fitted to their application. A first approach, we will not focus on, is the numerical modeling of fractured porous rock through a continuous numerical modeling. The numerical approaches are in these cases similar to the one used for the continuous medium, with additional functions to establish correlations, for example, fracturing pressure functions, porosity
stress, or permeability
stress relationships (Huang et al., 2015). Such an approach is not efficient when dealing with porous media where dominant fractures rule the physics of the system. We will rather focus here to the methods that explicitly account for the presence of crack(s).

12.3.1.1 Interface element methods Watanabe et al. (2012) highlight that coupling FEM with interface elements (IEs) is a widespread approach in geotechnics. In this approach, the fracture is seen as a solid entity with or without thickness. The IEs have been applied to coupled HM or THM problems with preexisting fractures and Watanabe et al. give a brief review of such works. Two problems are underlined by Watanabe et al. when considering IEs: a complex meshing procedure essentially because the mesh must precisely represent a FE approximation field and numerical difficulties to solve the mechanics-related coupled problem because IEs are required for the mechanical resolution, but not for the hydraulic or thermal resolution. Watanabe et al. proposed an approach based on a XFEM interpolation around the crack in order to account for HM couplings in a fractured porous medium with IEs. They developed a lower dimensional IE with local enrichment approximations used in XFEM approach. The enrichment only concerns the displacement, not the pressure. Several cracks, with intersections, can be modeled, but only on preexisting fractures. The authors did not dealt with fracture propagation which allowed them to use an existing FE code and facilitated the coupling between mechanical and hydraulic processes resolution. Moreover, it was thus possible to make a standard FE numerical integration of the weak form of the equations, then not requiring the complex development linked to the XFEM implementation. The spatial discretization is of the Galerkin FEM type and the primary variables are displacement, pressure, and fracture relative displacement. Classical 2D finite elements are used for the porous media while linear elements are used for discrete discontinuities. To ensure the continuity of pressure, the linear elements coincide with the edges of the 2D elements of the matrix, and the nodes of both types of elements are shared. Time discretization is achieved through a first-order backward Euler finite difference scheme. In this approach, the discontinuity through the fracture is limited to the displacement, not to the pressure (or its gradient) and eventually only one fluid pressure is considered, the pressure in the fracture being considered as part of the pressure in the porous medium. As a consequence, the flows between fracture and matrix are not calculated.

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Carrier and Granet (2012) developed and implemented (in Code_Aster, Edf) a new hydromechanical zero-thickness IE in the framework of finite element modeling of fluid-driven fracture propagation. The element can be combined to classical finite element for the rock matrix. The IE accounts for fluid flow in the fracture and exchange of fluids between fracture and porous medium, as well as fracture propagation on a predefined path, on the basis of a regularized cohesive zone model (CZM). The CZM avoids the stress singularity at the crack tip which is an issue of the Griffith fracture theory often considered in the models. The element is based on an 8-nodes quadrangle. The interpolation of the displacement is quadratic while the interpolation of pore pressure is linear, to be consistent with the classical finite element implemented in the FE code. An additional degree of freedom is added, namely fracture pressure, also based on a linear interpolation. A Lagrange multiplier is also added as a degree of freedom, with a linear interpolation, to enforce the constraint of equality of pore pressure across the interface (weak discontinuity). The discrete formulation is based on an explicit finite difference scheme and the nonlinear matrix system is solved with a Newton
Raphson algorithm. Pouya (2015) presents a numerical method for coupled fluid flow in porous fractured rock based on FEM. The model accounts for interesting and numerous physical features like multiple, curved and intersecting fractures, fluid mass exchanges between fractures, and between fracture and porous matrix as well as partial saturation. Both flows in the matrix and in the cracks are considered, and mass balance is written on cracks intersections. On the mechanical point of view, cracks are seen as interfaces which are assigned a Mohr
Coulomb yield criterion. On these discontinuities, joints elements are considered and allow displacement jumps. The model is implemented in a FE code, and the resolution is based on two meshes, one for the hydraulic problem, and the other for the mechanical problem. The difference between both meshes lies in the splitting of nodes of the interfaces in the mechanical mesh, while only one node is required for the hydraulic mesh. The use of two meshes results in a sequential resolution of the mechanical and hydraulic problems, nevertheless achieved in the same numerical code, which provides better computational performances than a sequential resolution with two distinct codes. The main limitation of the approach lies in the fact that the cracks must be predefined, propagation can only occur along predefined paths. This is sufficient for crack reactivation, but restrains the application of the method when considering new cracks creation.

12.3.1.2 Meshless methods An important class of methods when dealing with numerical modeling of crack propagation consists of the meshless methods. The reader interested on details on these methods in the general framework of solid mechanics is referred to, amongst others, the papers of Belytschko et al. (1996) and Nguyen et al. (2008). In the framework of HM couplings, Oliaei et al. (2014) proposed an element free Galerkin (EFG) method applied to the hydromechanical numerical modeling of hydraulic fracturing initiation and propagation processes in a saturated porous medium.

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In this EFG method, interpolation is based on nodes rather than elements, so that an arbitrary discrete crack path, avoiding mesh dependency, can be modeled. In the work of Olaei et al., the shape functions are based on moving least-squares (MLS) rather than point interpolation method (PIM), hence, shape functions are both consistent and compatible unlike PIM shape functions which are only compatible. The discrete system of equations is written under incremental constrained Galerkin weak form through a weighted residual method. A θ-scheme is used for time discretization and a penalty method for implementation of essential boundary conditions. Discrete fractures are modeled with the diffraction method providing shape functions that are continuous within influence domains, but discontinuous across the crack. As for Wang et al. (2002, 2010), they develop a meshless method based on radial PIM shape functions combined with a FEM within the framework of the Biot’s consolidation theory. Time integration scheme is once again a θ-scheme. Moreover, the model of Wang et al. (2010) accounts for crack-induced anisotropy, permeability evolution, and softening. A specific algorithm allows the coupling of meshless method and FEM: the crack zone is treated with the radial PIM meshless method, while the noncracked domain is treated with FEM. Unlike the work of Oliaei et al., discrete fractures are modeled with the visibility method.

12.3.1.3 Hybrid methods Partition of unity finite element method Wu and Wong (2014) use a numerical manifold method (NMM) to couple fluid flow and fracturing problems. NMM is a hybrid method coupling the FEM and the DDA. The NMM is a PUM. It makes use of local approximation functions for the elements fully cut by a crack and enriched local approximation functions for the elements partially cut by a crack. The set of PDEs is written under integral, then weak form. The space integration is achieved through a simplex-Gauss integration method for the enriched elements, and through a simplex method for nonenriched elements. The Newmark’s integration scheme is used for time integration. Wu and Wong adapt the NMM to account for HM coupling. The model is limited to fluid flow along with fractures contained between impermeable blocks, so that no fluid exchange is accounted for between fracture and matrix. A cubic law is adopted to relate fracture permeability to its aperture, so that the fracture conductivity depends on mechanical deformation. In return, water pressure is translated to a nodal load applied on the fracture lips, and a change in load induces a change of aperture through the normal stiffness. The crack propagation direction is determined with the criterion of maximum tensile hoop stress due to Erdogan and Sih. Due to crack propagation, the manifold elements and the physical covers (meshes) have to be updated. The model is mainly applied to analysis of rock slope failure. Remij et al. (2015) propose a PUM they called enhanced local pressure which can be seen as a PUFEM method. In this approach, not only the displacement field across the fracture is written as a strong discontinuity as in XFEM, but also the pressure field. To this aim, displacement and pressure fields are enriched by additional degrees of freedom for the nodes surrounding the crack. In addition, the

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authors introduce a distinct degree of freedom for fluid pressure in the fracture. In this approach, high pressure gradients near the cracks can also be modeled. The mechanical behavior of the fracture is described with a cohesive law, which relates the traction at the crack lip to the displacement jump across the crack, and allows to model the propagation of the crack. The model can then be applied to the simulation of fluid-driven fracture in porous rocks.

Combination of numerical methods Zhao et al. (2011) made numerical calculations to study the effects of stress on solute transport in cracked rocks, with a hybrid approach. The authors combine a DEM approach for the resolution of the coupled stress-flow problem in a given fracture network, while the transport simulations are achieved with a particle tracking method, which require a chaining between two numerical codes. One important limitation of the method is that the rock matrix is assumed to be impermeable, and the fluid conduction is limited to connected fractures. Geiger et al. (2004) coupled FEM and FVM for numerical modeling of multiphase flow, which results in a finite element finite volume method (FEFVM). They underline some of the limitations of the pure coupled FEM: poorly conditioned and not diagonally dominant matrices impeding the use of fast matrix solvers and nonlinearities requiring numerical heavy iterative schemes. The FEFVM method is based on the construction of a FEM grid and a FVM subgrid. FEFVM seems to be more accurate and less time consuming than fully coupled FEM. Nevertheless, Geiger et al. do not account for mechanics in their works, so that we will not insist here on this approach. FEFVM is also considered, together with FEMDEM (combination of FEM and DEM) by Latham et al. (2013) and Lei et al. (2014). Lecampion and Desroches (2015) used a combination of a FVM for the resolution of a fluid flow problem and the displacement discontinuity (DD) method for coupled mechanical model. They applied their approach to the modeling of initiation and propagation of several parallel radial hydraulic fractures, confined to mode I and in two dimensions. Crack initiation and propagation are based on linear elastic fracture mechanics (LEFM) and stress intensity Factor and the resolution of nonlinear set of equations is achieved with Newton implicit algorithm.

12.3.1.4 Boundary elements method Fidelibus (2007) developed an approach coupling BEM, for resolution of mechanical and fluid diffusion in poroelastic blocks, and FEM, to treat the diffusion problem in the fracture. The aim of the method is to obtain a good accuracy while maintaining reasonable computational cost. The poromechanical model considered for the poroelastic blocks is based on Biot theory and results in an equation for mechanics and an equation for diffusion, which are coupled. Considering a BEM scheme, these equations are transformed to exact integral equations only defined at the boundaries. The boundaries are then discretized in three-node elements to bring back integral equations to algebraic equations. A continuity equation is written for the diffusion problem in the fractures, on which is applied a FEM method with

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Galerkin weighted residuals before discretization. The fracture discretization is the same as the block faces discretization used for the BEM, with straight elements. For both blocks and fractures discretization, constant elements are considered for displacements, traction, and discharges, that is, displacements, tractions and discharges are uniform in a given element, while quadratic elements are considered for fluid pressure. Fidelibus underlines that its approach gives fair solutions considering at the same time a coarse mesh and relatively large time steps, thanks to the enhanced precision of the BEM scheme. Moreover, the approach saves computational time and spares the numerical modeling from instability problems. Ghassemi et al. (2013) also combined the FEM and the BEM to solve the coupled fracture flow and rock deformation problem to analyze the rock failure around a hydraulic fracture in 3D. The FEM solves the fluid flow in fracture while the BEM solves the solid mechanics part of the problem. The stress and pore pressure fields calculated around the main fracture are then used in a failure criterion to infer the potential for rock failure. In the approach of Ghassemi et al., the BEM is based on the DD formulation and sparses the authors from modeling the whole reservoir. The model accounts for leak-off from crack to matrix and allows the assessment of pore pressure and stress fields in the formation around a hydraulic fracture as well as the aperture of the hydraulic fracture. From the stress state, a Mohr
Coulomb failure criterion is used to assess the failure zones around the hydraulic fracture. Rawal et al. (2014) extended the same approach to coupled thermoporochemomechanical processes, in the framework of water injection and reactive flow into an enhanced geothermal reservoir. Reactive solute transport (fluid flow, heat, and solute transport) in the fracture is modeled with FEM, while 3D fluid diffusion, heat conduction and mineral diffusion problem in the rock matrix is modeled with BEM. The authors use the SUPG method for the resolution of the FEM problem to avoid numerical oscillations. The method allows the assessment of mineral concentration in the fracture and fracture aperture, as well as pressure, temperature, and stresses on the fracture plane and in the rock mass reservoir. The equations and details on the numerical implementation are provided in the papers of Ghassemi et al. (2013) and Rawal et al. (2014). Zhang et al. (2007, 2009) developed a numerical model to treat fluid-driven fracture propagation in fractured reservoirs. The mathematical model accounts for and couples fluid flow, elastic deformation, and frictional sliding. Fracture opening is controlled by both fluid pressure exceeding the normal stress acting on them in mode I and interactions with intersecting closed fractures ruled by Coulomb type frictional slip in mode II. The Newtonian fluid is supposed to flow through the conductive fractures according to a cubic lubrication equation relating the fracture conductivity to an equivalent hydraulic aperture. The rock material is water tight and follows an elastic constitutive law. The fracture propagation direction is given by the criterion of maximum tensile hoop stress due to Erdogan and Sih. The same model was applied by Zhang and Jeffrey (2014) to fluid-driven fractures in forming fractures networks in 2D plain strain, in order to simulate the propagation of hydraulic fractures through a fracture network, accounting for fluid-driven crack nucleation, propagation, and coalescence. A limitation of the model lies in the fact

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that the rock mass is supposed to be impermeable with fluid flow limited along the fractures. As a consequence, no exchange between crack and matrix fluid is accounted for. Zhang et al. (2015) extend the model to thermal solicitations. On a numerical point of view, the problem is solved by coupling a BEM based on a DD formulation, as for Ghassemi et al., to treat the mechanical problem in the rock mass, and a FDM to solve the fluid flow problem and assess the evolution of hydraulic aperture. To ensure numerical convergence, the time step is adjusted, while an adaptive meshing scheme is implemented to treat the propagation of fractures, in the direction of propagation, to ensure accuracy. The created elements near the crack tip are smaller in size than the standard DD elements, and the mesh is adjusted at each propagation step of the cracks.

12.3.1.5 Discrete element method Another important class of approaches is that based on DEM. Damjanac and Cundall (2016) present an approach based on DEM for the simulation of hydraulic fracturing accounting for HM couplings. A particularly interesting feature in their approach lies in the fact that it accounts for a preexisting discrete fracture network (DFN). DEM is able to treat coupled HM model for flow in cracked rock mass modeling. Propagation of crack along predefined paths can be accounted for by allowing progressive failure of the interfaces defined between blocks. Damjanac and Cundall give a review of the use of DFN in the framework of HM couplings in rock mass and fluid flow in deformable fractured rock mass. They present two application cases of DEM for hydraulic facture propagation. The first one consists in the validation of the model on the analytical solution provided by the PKN problem. The second example deals with a 3D modeling of hydraulic fracturing in a naturally fractured reservoir, which is a problematic of growing interest. This is a major advantage of the approach of Damjanac and Cundall, in which the DEM allows to account for a DFN with numerous fully or partially connected cracks. As already mentioned, the main limitation is the need for the propagating fluid-driven fracture to be path-predefined. The results of the simulations exhibit the interaction between hydraulic fracture and the DFN. Damjanac and Cundall then developed a so-called synthetic rock mass (SRM) approach to represent the HM behavior of a fractured rock mass. SRM is based on a one hand on a bonded particle model (BPM) to represent the deformation and damage of intact rock by DEM, and on the other hand on a smooth joint model (SJM) to treat the mechanical behavior of preexisting discontinuities, described by a 3D DFN. Due to the difficulty to represent deterministically the whole real DFN, statistically equivalent synthetic DFN are used in the SJM. The calibration of each component of the SRM can be achieved through laboratory tests. In the initially intact blocks, ruled by the BPM, the springs between nodes can break when their strength is exceeded, resulting in the creation of microcracks that can coalesce to form macrocracks. The flow in the fracture is based, quite classically, on the lubrication equation with a cubic law relating the flow within the crack to its aperture. The fluid flow in the intact porous rock blocks can also be accounted for in the SRM approach. The HM couplings are

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essentially based on the fact that fluid pressure influences mechanical deformation and strength of the rock, with consideration of an effective stress concept, while the mechanical deformation, as a back effect, influences fluid pressures and crack aperture, thus crack permeability. The coupling scheme is based on a relaxation parameter, which is a function of the rock bulk modulus and the lattice resolution. The time integration scheme is explicit. The approach is applied to the modeling of the propagation of a hydraulic fracture and its interaction with preexisting DFN.

12.3.1.6 Extended finite element method XFEM methods can be viewed as a variation of the PUFEM methods. XFEM method can deal with both strong and weak discontinuities. With XFEM method, remeshing is not required since the geometry of the fracture does not depend on mesh structure, that is to say the fracture is not meshed. Instead, discontinuous enrichment functions are introduced to account for primary variables discontinuity (strong discontinuity) or primary variables gradient discontinuity (weak discontinuity). This enrichment is made locally, which allows a reduction of the calculation time dedicated to rigidity matrices inversion and limits the number of additional unknowns when compared to most of meshless methods based on a global enrichment. A classically used enrichment function is the Heaviside function, which allows to model the discontinuity of displacement field in a thin band around the crack. Different types of enrichment are considered depending on whether the element is fully crossed by a crack or not. Thus, at the crack tip, singular functions are introduced to account for the asymptotic behavior at the crack tip. These different enrichments result in additional degrees of freedom for the nodes surrounding the cracks. In addition, the criterion for crack propagation is in most cases based on the stress intensity factors at the crack tip, in the framework of LEFM, which allows to express the crack tip enrichment functions, depending on the failure mode. The approximation of the displacement field in XFEM can be written: uð x Þ 5

X iAE

ϕi ðxÞai 1

X jAF

ϕj ðxÞHbj 1

X kAG

ϕk ðxÞ

4 X α51

eαk Fα

In this expression, F is the set of nodes of the elements fully crossed by the crack, G is the set of nodes of the elements containing the crack tip (Fig. 12.1), H is the generalized Heaviside function, and Fα is the crack tip enrichment functions. H and Fα are respectively associated to the additional degrees of freedom bj and ek. 





 pffiffi pffiffi pffiffi pffiffi θr θr θr θr r sin r cos Fα 5 ; r cos ; r sin sinðθr Þ; sinðθr Þ 2 2 2 2 with (r,θr) the local basis at the crack tip. Recent developments, in the context of hydraulic fracture, attempted to account for a full HM coupling, and were based on CZMs. LEFM models, indeed, allow to

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Figure 12.1 Representation of a crack nonconform to the mesh with nodes enriched by Heaviside function (dot) and crack tip functions (squares).

predict the hydraulic fracture behavior in brittle hard rocks, but are less efficient and accurate for crack propagation in ductile rocks (Wang, 2015). XFEM methods were first proposed by Belytschko et al. (1996); Belytschko and Black (1999) and Moe¨s et al. (1999) for solid mechanics problems, without accounting for HM couplings. XFEM is a widespread method to treat the propagation of cracks in rocks, when focusing on mechanical behaviors of both crack and matrix. Nevertheless, its extension to coupled fluid flow and hydraulic fracture propagation is more recent, and a number of studies are limited to not fully coupled pressure diffusion and rock deformation problems. XFEM method has recently been combined with HM coupling formalism by several authors. Khoei and Haghighat (2011) modeled the behavior of a fractured fully saturated porous medium with XFEM, without considering any crack propagation. They considered weak interfaces and used level set distance function as enrichment function. Khoei et al. (2012) extended the model to account for THM couplings in fully saturated porous medium, considering strong discontinuities for displacement, pressure and temperature fields across an impermeable fracture. At the fracture tip, they used

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crack tip asymptotic functions for the displacement field and simplified asymptotic functions for pressure and temperature fields. No propagation was considered. Mohammadnejad and Khoei (2013b) introduced crack propagation in the model through a CZM. A strong discontinuity is applied to the displacement field, while a weak discontinuity is applied to the pressure field, allowing the latter to be continuous across the fracture and allowing fluid leak-off from the fracture. Mohammadnejad and Khoei (2013a) extended the model to the partially saturated domain in the case of a weak discontinuity (bimaterial) on displacement. Several enrichment functions are tested on the displacement field. There is no discontinuity on both fluid and capillary pressures, neither strong nor weak, so that no enrichment is applied to these primary fields which are thus approximated with the classical FEM. The authors compare the efficiency, in terms on convergence of four different enrichment functions, on the results of displacements and pressure fields. They showed that two different approaches gave improved accuracy and convergence, which they called “standard XFEM with modified abs-enrichment” and “corrected XFEM”. The former makes use of enrichment functions proposed by Moes et al. (2003), while the latter is due to Fries (2008). As an illustration, Mohammadnejad and Khoei (2013a) give, amongst others, the comparison of the convergence results on a problem of water drainage from a biomaterial sand column, which show that “standard XFEM with modified abs-enrichment” and “corrected XFEM” provide same accuracy and convergence results. Mohammadnejad and Khoei (2013c) considered the crack propagation with HM couplings in partially saturated medium with XFEM and a CZM. A strong discontinuity is taken into account for the displacement field to account for the jump across the fracture, while a weak discontinuity is accounted for the fluid and capillary pressures, preventing them from being discontinuous while fluids flow jumps perpendicular to the fracture are allowed. Gordeliy and Peirce (2013a, 2013b) used the XFEM method to study propagation of hydraulic fractures. Wang (2015) developed a XFEM based model aiming at accounting for nonplanar hydraulic fractures, CZM, and nonelastic behaviors. The fluid flow in the fracture is described by a cubic law that relates fluid flow to pressure gradient as a function of fracture aperture. Fluid exchange between fracture and matrix is considered. The coupling between hydraulic and mechanical behavior of the rock matrix is based on Biot theory. The CZM is a traction-separation law. The combination of XFEM and CZM allows to propagate the fracture in any direction without predefining the crack path. Another consequence of the use of CZM is that it is not required to consider the enrichment at the crack tip. One can only consider the enrichment for the element fully crossed by the crack. In the model, no enrichment seems to have been taken into account for the pressure field, so that the pore pressure is continuous. The HM coupling between fracture and matrix mainly lies in the fluid leak-off from fracture to the rock matrix. Prevost and Sukumar (2016) used an XFEM method combined with HM couplings in 3D. A strong discontinuity is considered for the displacement field across the crack, while two formulations are considered for the pressure field depending on the geological problem modeled: either a weak discontinuity, with a continuous pressure field but a discontinuous pressure gradient field, to model a

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flow conduit, or a strong discontinuity, with a discontinuous pressure field, to model a sealing fault. Slipping is allowed along the crack, according to a Mohr
Coulomb yield criterion. No propagation of crack is accounted for in this model, but several, not intersecting cracks, can be considered.

12.3.2 A focus on an XFEM approach for numerical modeling of fully saturated fractured rock mass with HM couplings In this section, we present a more detailed model developed by Faivre et al. (2016) in order to highlight some numerical issues linked to the modeling of fluid-driven cracks in porous rocks. This model is developed under an XFEM approach in the framework of hydromechanical couplings for a fully saturated porous medium.

12.3.2.1 Physical model Let us consider a domain Ω C R2 entirely cut by a discontinuity Γc. The porous medium is supposed to be fully saturated with a single-fluid phase and fluid flow occurs in the crack. Thus, a pressure p is considered for the interstitial fluid in the porous matrix and another pressure pf is considered for the fluid in the fracture. u is the displacement field and u the displacement jump (at the crack interface). The pressure discontinuity is written like a strong discontinuity. Nevertheless, a boundary condition is imposed on the fracture wall that enforces the pressure in the fracture pf to be equal to pore pressure in the matrix p on the fracture wall, thus ensuring a weak discontinuity on fluid pressure. As a consequence, only the pore pressure gradient field is discontinuous. Two Lagrange multipliers are introduced to meet this condition on fracture walls, which are interpreted as potential mass interfacial flows between the surrounding porous medium and the crack. The main equations of the model are the mass balance equation for both the pore fluid pressure p and the fracture fluid pressure pf. The mass balance equation for the fracture fluid is a lubrication equation that accounts for mass interfacial fluxes occurring between the fracture and the surrounding porous medium. Darcy’s law rules the mass flux in the rock matrix, while a cubic law rules the fluid flow inside the fracture, as a function of fracture aperture. Variations of fluid density in both the matrix and the fracture are considered, as well as variations of porosity of the matrix. In this model, the authors use a CZM for the propagation of the fracture, along a path that is predefined. Thus, the fracture is decomposed into three zones: G

G

G

an entirely opened zone where the total stress on the fracture walls is only due to the fluid; a partially opened zone (fracture process zone); a nondamaged zone where the two leaps of the cracks are in perfect adhesion with no interpenetration and fluid circulation is forbidden.

The fracture mentioned above is entirely invaded by the fluid. Moreover, the CZM is based on the minimization of the total potential energy, which is a function of the displacement field u and of the fracture aperture δ 5 ½u.

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12.3.2.2 Numerical implementation and numerical scheme Cohesive zone model Amongst the family of CZM, two classes can be distinguished: regularized CZM and nonregularized CZM. The former require a penalization parameter and Ferte´ et al. (2016) showed, on the basis of an inclusion debonding test, that a too small penalization parameter induces a fracture aperture too far from the reference solution given in FEM with conforming meshes, exhibiting nonphysical opening or interpenetration. A too high value of the penalization parameter results in spurious oscillations of the traction in the adherent zone, due to the violation of the inf-sup condition. On the contrary, the nonregularized CZM satisfy the inf-sup condition, especially considering specific set of active constraints on the Lagrange multipliers. Thus, the HM-XFEM model proposed by Faivre et al. (2016) is based on a nonregularized CZM. The cohesive stress is given by the Talon
Curnier constitutive law, which is written here under the assumption of effective stress. The numerical formulation of the nonregularized CZM requires the introduction of two additional degrees of freedom. Indeed, the field δ, representing the fracture aperture, is considered as an unknown of the mathematical formulation, while a field λ is added to enforce the constraint δ 5 ½u in an augmented Lagrangian.

Fields interpolation In the proposed model, the pore pressure field is weakly discontinuous, that is to say pore pressure field is continuous through the fracture, with the continuity condition p 5 pf, while its gradient is discontinuous. The XFEM enrichment considered in the model allows a strong discontinuity of the pore pressure field across the fracture, but pore pressure continuity is weakly enforced at each crack lip via Lagrange multipliers. Thus, the approximation of the pore pressure field is similar to that of displacement and is given by: pð x Þ 5

X iAIs

ci ψi ðxÞ 1

X

dj ψj ðxÞH ðlsnðxÞÞ

jAJs

As already mentioned, the standard finite element approximation is enriched around the crack with an enrichment function associated to the additional degrees of freedom. Is is the set of vertex nodes whose support contains x, while Js is the set of vertex nodes whose support is fully crossed by the crack. Thus, unlike the displacement field, only the vertex nodes are enriched for the pore pressure field. This results in a linear interpolation of this pore pressure field, while the interpolation of the displacement field is quadratic. This must be linked to the order of interpolation of the classic finite element, for which we have already mentioned in part 2 that considering different orders of interpolation allows to reduce oscillations of the solution when highly transient boundary conditions are imposed. In the Heaviside function, lsn(x) refers to the normal level set which allows to describe the crack geometrically. Two level-sets can be considered: normal level set whose iso-zero describes the fracture surface and tangential level set whose iso-zero allows

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to locate the crack front. Depending on the interpolation considered for the normal level set, either straight interfaces (linear shape functions) or curved interfaces (quadratic shape functions) can be modeled. Due to the enrichment at the nodes around the crack, there exist several kinds of elements in the matrix. The so-called reproducing elements are fully crossed by the cracks for which all the nodes are enriched, while the so-called blending elements are adjacent to the reproducing elements and have both enriched and nonenriched nodes. The latter do not fulfill the partition of unity and thus highly decrease the convergence rate of the numerical solution. The choice of the generalized Heaviside functions as enrichment functions allows to address this issue. Moreover, the unnecessary enriched degrees of freedom of the blending element are deleted by setting them to zero, which permits the presence of both blending and standard finite elements. Along the fracture, all the fields are continuous since both sides of the fracture are represented with a single line, which is modeled by adding virtual 1D quadratic elements on the basis of the normal level set. As a consequence no enrichment is taken into account for these fields. A set of intersection points between the crack and the reproducing elements is defined, on which is defined the field δ. The degrees of freedom corresponding to pf and Lagrange multipliers (enforcing pore pressure continuity and equality of δ and u) are added to the vertex nodes of the reproducing elements. The dimensions of the Lagrange multipliers space are chosen to ensure a good convergence rate and prevent the solution from numerical oscillations. The method is based on the notion of vital edges, and the reader is referred to the paper of Faivre et al. (2016) for more details.

12.4

Conclusions and prospects

A variety of numerical methods have been used to model coupled fluid flow in porous rocks. Most of these methods are adapted from methods developed in the areas of solid or fluid mechanics. Even if various physical phenomena have been modeled with success, such as crack propagation, both on predefined and nonpredefined paths, curved cracks or crack intersection, most of the case in the saturated case, some challenges remain in this research field. For example, when dealing with porous rocks, as for most rocks, a paramount feature is the structural anisotropy of the material and its modeling. To the authors’ knowledge, few works have dealt with coupled fluid flow in fractured porous anisotropic rocks so far. Crack initiation also still needs some particular attention. An important feature is the computing requirements (time, memory) on these methods, and development of parallelizable methods and softwares is critical for efficiency of the calculations.

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True triaxial failure stress and failure plane of two porous sandstones subjected to two distinct loading paths

13

Xiaodong Ma1,2, Bezalel C. Haimson1 and John W. Rudnicki3 1 University of Wisconsin-Madison, Madison, WI, United States, 2Stanford University, Stanford, CA, United States, 3Northwestern University, Evanston, IL, United States

13.1

Introduction

The objective of this research was to elucidate failure mechanisms in porous sandstones when subjected to a general compressive state of stress (σ1 $ σ2 $ σ3 $ 0). Porous sandstone formations can host oil and gas reservoirs and form waste storage sites (Wawersik et al., 2001). Activities related to hydrocarbon exploitation or waste disposal are commonly associated with the failure of the host sandstones, either by design or by accident. Except for the rare case in which a combination of very weak rock and very high stresses brings about cataclastic flow, failure in porous sandstones is exhibited by the development of localized, tabular zones that we call “failure planes” (Fig. 13.1). These planes take the form of shear bands or faults at a steep angle θ (where θ is the angle between the normal to the plane and the σ1 direction) under lower prevailing in situ stresses, or compaction bands (perpendicular to the σ1 direction, i.e., θ 5 0
) under higher stress regimes. Extensive laboratory work has been dedicated to characterizing failure in porous sandstones (Wong et al., 1997; Be´suelle et al., 2000; Mair et al., 2002; Wong and Baud, 2012). The information provided by these investigations was largely derived from conventional triaxial compression tests. Some tests in sandstones were conducted under triaxial extension (σ1 5 σ2 . σ3) (Zhu et al., 1997; Be´suelle et al., 2000; Bobich, 2005). However, field evidences revealed that the stress condition in the Earth’s crust is rarely axisymmetric (as being simulated in conventional triaxial tests) (Haimson, 1978; McGarr and Gay, 1978; Brace and Kohlstedt, 1980; Vernik and Zoback, 1992). The knowledge derived from conventional triaxial tests was inadequate for the understanding of failure under typical crustal stress conditions where σ1 . σ2 . σ3. Mogi (1971) pioneered a new method of testing mechanical behavior rocks subjected to three unequal principal stresses (true triaxial testing). Since then, several series of true triaxial tests have been reported (Mogi, 1971; Michelis, 1985; Porous Rock Failure Mechanics. DOI: http://dx.doi.org/10.1016/B978-0-08-100781-5.00013-0 © 2017 Elsevier Ltd. All rights reserved.

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σ1

θ

σ2

σ3 θ

Figure 13.1 Illustration of the failure plane and the measurement of failure-plane angle (θ). Modified from Ma and Haimson (2016).

Takahashi and Koide, 1989; Wawersik et al., 1997; Haimson and Chang, 2000; Young et al., 2012). These tests have revealed the consistent effect of σ2 on rock failure, deformability, and failure-plane angle. However, until very recently, only a few true triaxial tests had been performed on porous sandstones (Takahashi and Koide, 1989; Wawersik et al., 1997; Oku et al., 2007; Ingraham et al., 2013; Ma and Haimson, 2016). These tests were conducted mostly on low-porosity (n , 10%) sandstones, which undergo little or no compactive deformation under high crustal stress magnitudes. In this contribution, exhaustive true triaxial tests in two quartz-rich, weakly cemented porous sandstones [Coconino sandstone (porosity 5 17.5%) and Bentheim sandstone (24%)] are summarized. Both rocks are representative of formations that tend to undergo compactive deformation (Lee, 2005; Klein et al., 2001; Vajdova et al., 2004; Stanchits et al., 2009). In the first phase of this study, the effect of σ2 on failure, failure-plane angle, and failure mode was studied using a common loading path (Ma and Haimson, 2016). This loading path maintains σ2 and σ3 at preset magnitudes while raising σ1 monotonically until failure occurs. In the common loading path tests, the three principal stress invariants (the octahedral shear and normal stresses, τ oct and σoct, and the Lode angle Θ) constantly change during loading. To facilitate comparison of experimental failure characteristics with existing theories, a novel loading path was designed to maintain a fixed Θ during the test. The second phase of this research comprised of novel loading path tests (Ma et al., in revision) aimed at characterizing the dependence of failure and failure-plane angle on σoct and Θ. In the third phase of this study [for details, see Ma (2014)], experimental results on failure and failure-plane angle were compared with the localization theory (Rudnicki and Rice, 1975), which describes the failure-plane angle θ based on its dependence on σoct and Θ, as constrained by the novel loading path tests. The theoretical angle model was first verified with the measured angles in the novel loading

True triaxial failure stress and failure plane of two porous sandstones

287

path experiments. Then, it was extended to predict the variation of θ with σ2 under constant σ3 for a comparison with observations in the common loading path tests.

13.2

Materials

Two quartz-rich porous sandstones were selected for this study: Coconino and Bentheim (Fig. 13.2). Both sandstones are deemed representative of formations that tend to undergo compactive failure when subjected to high compressive stresses. Coconino sandstone is of Permian age, has a medium to high porosity, composed of 96% quartz, 2% feldspar, and 2% iron oxide and clay (Lee, 2005). Its fine grains (0.10 6 0.03 mm) are well sorted and rounded. The block used in this study was obtained from a quarry in northern Arizona. It is pink with dark, fine (about 1 mm thickness) bedding planes. It had been tested by Friedman and Logan (1973) under conventional triaxial stresses. It was also used by Lee (2005) in drilling experiments leading to tabular borehole breakouts when subjected to high true-triaxial stress conditions. This feature suggested that these breakouts developed as a result of the formation of compaction bands ahead of the breakout tip. Bentheim sandstone is Lower Cretaceous from Bentheim, Germany. Similar to Coconino sandstone, it is also quartz rich, but has a significantly higher porosity. Bentheim sandstone is composed of 95% quartz, 3% feldspar, and 2% kaolinite (Klein and Reuschle´, 2003). The mean grain size is about 0.3 mm. Its quartz grains are rounded to subrounded. The block from which test specimens was cut for this study is yellow to light-brown, bearing hardly visible bedding planes. Bentheim sandstone has been tested extensively under conventional triaxial stresses. In all tests, compaction bands were observed at high stress levels (Klein et al., 2001; Klein and Reuschle´, 2003; Vajdova et al., 2004; Stanchits et al., 2009). Ma (2014) measured the uniaxial deformational and failure characteristics parallel and perpendicular to the bedding planes in Coconino and Bentheim sandstones. Although both rocks are bedded sandstones, insignificant discrepancies (typically less than 5%) were found between these two directions. For consistency, all tested specimens were prepared so that their long axis was perpendicular to bedding.

Figure 13.2 SEM micrographs of untested specimen of Coconino (left) and Bentheim (right) sandstone. From Ma and Haimson (2016).

288

13.3

Porous Rock Failure Mechanics

True triaxial experiment procedures

The University of Wisconsin true triaxial testing assembly (for details, see Haimson and Chang, 2000) was employed to conduct experiments in this study. The assembly consists of a cubical true triaxial cell housed inside a cylindrical biaxial loading apparatus (Fig. 13.3). The true triaxial cell accommodates a 19 3 19 3 38.5 mm3 cuboidal rock specimen. Specimens were saw-cut and surface-ground into cuboids with a finish of 5 μm flatness on each face.

13.3.1 Common loading path The common loading path (Ma and Haimson, 2016) is composed of three stages (Fig. 13.4): 1. The specimen was loaded monotonically under hydrostatic condition (σ1 5 σ2 5 σ3), at a constant rate of 0.1 MPa/s, until the preset σ3 level was reached. 2. The σ3 was maintained constant while σ1 and σ2 were raised together monotonically at the same 1 MPa/s rate, until the predetermined σ2 was attained. 3. σ3 and σ2 were maintained constant while σ1 was monotonically increased until it reached a peak (σ1,peak), namely failure. σ1

(A)

f σ2

(B)

a

b

d e

c σ3

f

Figure 13.3 Schematic diagram of the true triaxial testing apparatus: (A) Cross section and (B) profile; (a) biaxial loading apparatus; (b) polyaxial cell; (c) loading pistons; (d) confining fluid; (e) metal spacers; (f) rock specimen. From Ma and Haimson (2016).

Common loading path

1, peak

Stress

Stress

True triaxial failure stress and failure plane of two porous sandstones

Novel loading path

289

1, peak

2

2

3

3

(A)

Time

(B)

Time

Figure 13.4 Two distinct loading paths of true triaxial testing: (A) common loading path (σ2 and σ3 constant, σ1 increasing monotonically) and (B) novel loading path (σ3 constant, σ2 and σ1 rising while maintaining constant stress ratio b [ 5 (σ2  σ3)/(σ1  σ3)]). Modified from Ma et al. (in press).

13.3.2 Novel loading path The novel loading path (Ma et al., in revision) was designed to maintain constant the deviatoric stress state (quantified by the Lode angle Θ). Here we define Lode angle as: ( ) ðσ1 2 2σ2 1 σ3 Þ  (13.1) Θ 5 tan 2 1
O3ðσ1  σ3 Þ when σ2 5 σ3, Θ 5 130
and when σ2 5 σ1, Θ 5 230
. As illustrated in Fig. 13.4, loading begins with applying all three principal stresses hydrostatically up to the desired level, and then maintaining σ3 constant while raising σ1 and σ2 simultaneously at a fixed stress ratio b (5Δσ2/Δσ1) (Jimenez and Ma, 2013), where Δσ1 5 σ1 2 σ3 and Δσ2 5 σ2 2 σ3. The novel loading path differs from the common loading path in the way σ2 is controlled. The novel loading path maintains a constant Lode angle. The Lode angle Θ (Eq. 13.1), can be conveniently expressed in terms of the stress ratio b: " # ð 1  2b Þ Θ 5 tan21 (13.2) O3 When b 5 0, and Θ 5 130
, corresponding to axisymmetric compression case (σ2 5 σ3); when b 5 1/2, and Θ 5 0
, and deviatoric pure shear state [σ2 5 (σ1 1 σ3)/2] is achieved; and when b 5 1, and Θ 5 230
, simulating the axisymmetric extension state (σ2 5 σ1). The relationship between the stress ratios and the corresponding Lode angle is given in Table 13.1.

13.3.3 Post-test procedure After failure, all three principal stresses were simultaneously lowered to zero, while maintaining the relative true triaxial stress state condition, in order to minimize the

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Table 13.1 Relations between deviatoric stress state parameter N, stress ratio b, and Lode angle Θ Stress ratio, b

Lode angle Θ

N

Deviatoric stress state

1:1 3:4 1:2 1:3 1:6 0

230
216
0 111
121
130


21 20.58 0 0.37 0.71 1

Axisymmetric extension  Deviatoric pure shear   Axisymmetric compression

impact of unloading on rock damage. At the conclusion of each test, the specimen was extracted from the true triaxial cell, and the polyurethane cover on the σ3 faces was carefully removed. The specimen was first photographed, and then impregnated with a thin, penetrative epoxy to preserve failure conditions.

13.3.4 Testing program Two suites of true triaxial test series leading to compressive failure were conducted in each of the two sandstones. Specifically, the applied constant σ3 levels were 0, 10, 20, 50, 80, 100, 120, 150 MPa for Coconino sandstone, and 0, 8, 15, 30, 60, 80, 120, 150 MPa for Bentheim sandstone. In the common loading path tests, the desired magnitude of σ2 varied from test to test for each constant σ3, covering its entire range from σ2 5 σ3 to σ2 5 σ1. In the novel loading path tests, different constant Lode angle Θ values were applied for each constant σ3 magnitudes. In Coconino sandstone the applied constant Lode angle Θ values were 130
, 121
, 111
, 0
, and 230
. For Bentheim sandstone, the applied constant Lode angle Θ values were 130
, 121
, 111
, 0
, 216
, and 230
.

13.4

True triaxial failure under common loading path

13.4.1 True triaxial failure stress in terms of principal stresses All true triaxial stress conditions at failure are displayed in terms of σ1,peak as a function of σ2 for each constant σ3 (Fig. 13.5). For a given σ3, σ1,peak generally increases with σ2 until a maximum is reached at some σ2 value, beyond which σ1,peak gradually declines. For both sandstones, when σ2 approaches the magnitude of σ1, σ1,peak is approximately equal to its base value when σ2 5 σ3. Fig. 13.5 demonstrates that σ1,peak is a function of not only σ3, but also of σ2, as observed earlier in other rock types (Mogi, 2007; Haimson, 2006; Oku et al., 2007; Lee and Haimson, 2011). Generally, the strengthening effect of σ2 in both sandstones is appreciably less than in previously tested crystalline and low-porosity clastic rocks (Haimson and Chang, 2000; Chang and Haimson, 2000; Oku et al., 2007; Lee and Haimson, 2011).

True triaxial failure stress and failure plane of two porous sandstones

400

700 600 σ2 = σ3

150 120 100

500 σ1, peak (MPa)

291

σ2 = σ3

150 120 80 60

300

80

250

50 σ2 = σ1

200

400 300

350

σ2 = σ1 30

150 20

200 100 0

15

100

10

8 50

σ3 = 0 MPa

σ3 = 0 MPa

Coconino

Bentheim

0 0

200

400

600

0

100

σ2 (MPa)

200

300

400

σ2 (MPa)

Figure 13.5 Variation of σ1,peak with σ2 in Coconino and Bentheim sandstone tested under the common loading path for all constant σ3 levels: the cyan curve represents the Mohr strength criterion when σ2 5 σ3 and the magenta straight line connects data points of σ2 5 σ1. From Ma and Haimson (2016)

For example, in Westerly granite, σ1,peak increased by as much as 59% when σ3 5 20 MPa (Haimson and Chang, 2000); in 5% porosity TCDP (Taiwan Chelungpu-fault Drilling Project) sandstone the maximum σ1,peak increase was 77% at σ3 5 10 MPa (Oku et al., 2007). However, it is not conclusive whether the subsidence of σ2 strengthening effect is related to the porosity increase.

13.4.2 Failure mode and failure-plane angle Failure in both sandstones generally culminates in faulting or the development of shear bands or even compaction bands when Bentheim sandstone subjected to high σ3. Under conventional triaxial compression stress condition (σ2 5 σ3), a decline of failure-plane angle with the rise in σ3 was readily observed. Photographs of the faces normal to σ2 in specimens tested at σ2 5 σ3, demonstrate this trend (Fig. 13.6). In Coconino sandstone, θ dropped from about 80
(at σ3 5 0) to about 50
at the highest σ3 applied (5150 MPa). As shown in Fig. 13.6, at σ3 5 80 MPa and higher, multiple parallel and often conjugate shear bands were created (θ . 45
), characteristic of brittleductile transition (Paterson and Wong, 2005, pp. 213217). In Bentheim sandstone, the variation of θ with σ3 between 0 and 150 MPa was more dramatic. θ dropped from about 80
(at σ3 5 0) to approximately 45
at σ3 5 100 MPa, when multiple parallel and conjugate failure-planes were observed. For higher σ3, failureplane angle continued to decrease and at σ3 5 150 reached an extreme, θ 5 0
(i.e., plane perpendicular to σ1). The low-angle (θ , 45
) failure-planes are unlikely to be induced by shear stress as the shear component diminishes with the decline in θ. It is

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σ1 80°

72°

68°

62°

60°

54°

56°

50° σ3

(A)

σ2 = σ3 = 0

10

20

50

80

68°

72°

120

150 MPa

Brittle-Ductile transition

Brittle 80°

100

60°

55°

43°

20°

~0°

σ3

(B)

σ2 = σ3 = 0

8

15 Brittle

30

60

100

120

150 MPa

Brittle-Ductile transition Early ductile zone

Figure 13.6 Photographs of Coconino (A) and Bentheim (B) sandstone specimens tested under conventional triaxial compression (σ2 5 σ3): the failure-planes shown on the samples’ σ1σ3 surfaces demonstrate the failure-plane angle decrease and the failure mode evolution with σ3, indicating the brittleductile transition. From Ma and Haimson (2016).

suspected that these failure-planes are in fact compaction bands, which signifies the rock entered its early ductile zone. Under true triaxial stress conditions (σ1 . σ2 . σ3), failure-plane in both sandstones unambiguously dips in the σ3 direction. As σ2 was raised beyond σ3, the measured failure-plane angle generally increased for each constant σ3 levels (Fig. 13.7). This trend can be approximated by a straight line with reasonable goodness of fit (generally R2 $ 0.6). The increase of θ in Bentheim sandstone was milder than observed in Coconino sandstone and was limited to about 10
. For low σ3 (530 MPa), the increase was 9
(from 57
at σ2 5 30 MPa to 66
at σ2 5 180 MPa); at σ3 5 80 MPa, the increase in θ at σ3 5 80 MPa was slightly less, from 45
at σ2 5 80 MPa to 53
at σ2 5 285 MPa. The extreme case was the test series of σ3 5 150 MPa, in which all specimens developed compaction bands (θ  0
), suggesting no σ2 dependence. Unlike in Coconino sandstone, the appearance of failure-planes in Bentheim sandstone seems less affected by rising σ2 at all constant σ3 series. As depicted in Fig. 13.7, the increase in failure-plane angle in both sandstones as a function of σ2 was generally consistent with but was not as prominent than that observed previously in crystalline rocks (up to about 20
, see Haimson and Chang, 2000; Chang and Haimson, 2000; Lee and Haimson, 2011). The failure-plane angle variation with σ2 for a constant σ3 revealed by the true triaxial tests contradicts the Coulomb assumption that failure-plane angle is a unique rock property. The results are also in disagreement with Mohr failure criterion, which allows changes in failure-plane angle, but only as a function of σ3.

True triaxial failure stress and failure plane of two porous sandstones

90

85

σ3 = 0 MPa

Failure-plane angle, θ (°)

80

Coconino

10

293

σ3 = 0 MPa

80

8

70 75

20

70

Bentheim

15 30 60

60

80

50

50

65

40 80 100 120

60 55

150

20 10 150

50 45

120

30

0 0

200 400 σ2 (MPa)

600

0

100

200 σ2 (MPa)

300

400

Figure 13.7 Variation of failure-plane angle θ with σ2 in Coconino and Bentheim sandstone tested under the common loading path for each constant σ3 level: the data points were linearly fitted. From Ma and Haimson (2016).

13.5

True triaxial failure under novel loading path

13.5.1 True triaxial failure stress in terms of principal stress invariants Fig. 13.8 shows the variation of the octahedral shear stress at failure (τ oct,f) with the octahedral normal stress at failure (σoct,f) in both sandstones for constant Lode angle. For all Θ values tested, τ oct,f first rose as σoct,f was increased, but the rate of increase consistently decreased. In Bentheim sandstone, τ oct,f reached a peak and began declining in the vicinity of σoct,f 5 200 MPa. The τ oct,f vs σoct,f curve varied considerably with Θ between 130
and 230
. The σoct,f magnitude when the peak of each constant Θ curve was reached gradually increased as Lode angle Θ declined from 130
to 230
. In other words, for a constant τ oct,f, the σoct,f magnitude required for failure was typically increased as Θ varied from 130
to 230
. The relationship of τ oct,f vs σoct,f for each constant Θ can be well fitted by a second-order polynomial equation. Fig. 13.8 demonstrates the consistent effect of Lode angle Θ on the dependency of τ oct,f on σoct,f. It reveals that the τ oct,fσoct,f relationship obtained from axisymmetric tests are special cases only, and that the τ oct,fσoct,f relationship based on common loading path tests without differentiating Lode angle Θ is a coarse approximation to all Θ conditions.

13.5.2 Failure-plane angle Failure-plane angles θ in both sandstones were plotted in Fig. 13.9 as a function of σoct,f, for each constant Lode angle. The angle θ decreased steadily with the

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120 Coconino

Bentheim

τoct,f (MPa)

200

100 80

150

60 Θ = +30°(b = 0) Θ = +21°(b = 1/6) Θ = +11° (b = 1/3) Θ = 0° (b = 1/2) Θ = –16° (b = 3/4) Θ = –30° (b = 1)

100 Θ = +30° (β = 0) Θ = +21° (β = 1/6) Θ = +11° (β = 1/3) Θ = 0° (β = 1/2) Θ = –30° (β = 1)

50

0 0

100

200

300

400

500

40 20 0

0

100

σoct,f (MPa)

200

300

σoct,f (MPa)

Figure 13.8 Variation of the octahedral shear stress at failure (τ oct,f) with the mean stress at failure (σoct,f) for all constant Lode angles in Coconino and Bentheim sandstones. From Ma et al. (in press; in revision).

85 Coconino

Bentheim

Failure-plane angle, θ (°)

80 80 75 60

70 65

40

60

Θ = +30° (b = 0) Θ = +21° (b = 1/6) Θ = +11° (b = 1/3) Θ = 0 (b = 1/2) Θ = –30° (b = 1)

55 50

20 0

45 0

100

200

300

σoct,f (MPa)

400

500

Θ = +30° (b = 0) Θ = +21° (b = 1/6) Θ = +11° (b = 1/3) Θ = 0° (b = 1/2) Θ = –16° (b = 3/4) Θ = –30° (b = 1)

0

100

200

300

σoct,f (MPa)

Figure 13.9 Variation of the failure-plane angle (θ) with the mean stress at failure (σoct,f) for all constant Lode angles in Coconino and Bentheim sandstones. From Ma et al. (in revision).

rise in σoct,f, from approximately 80
at the lowest σoct,f (  50 MPa) to about 50
in Coconino sandstone as σoct,f approached 350 MPa. In Bentheim sandstone, the failure-plane angle dropped monotonically from approximately 85
at the lowest σoct,f (  15 MPa) to about 0
as σoct,f approached 200 MPa. The variation of Θ with σoct,f for each test series conducted for constant Lode angle could be linearly fitted. For the same σoct,f, failure-plane angle increased by up to 16
as the Lode angle Θ varied from 130
to 230
(corresponding to the stress ratio b rising from 0 to 1).

True triaxial failure stress and failure plane of two porous sandstones

13.6

295

Prediction of failure-plane angle via bifurcation theory

The novel loading path tests in both Coconino and Bentheim sandstones have revealed that for a constant Lode angle Θ the failure-plane angle (θ) monotonically decreased with the rise in mean stress at failure σoct,f. On the other hand, the angle for a given σoct,f increased as the Lode angle Θ varied from 130
(axisymmetric compression) to 230
(axisymmetric extension). The angle dependence on both Θ and σoct,f was generally consistent with the shear localization theory introduced by Rudnicki and Rice (1975). In this section, a theoretical prediction of failure-plane angle based on a two-invariant failure criterion (Drucker and Prager, 1952) proposed by Rudnicki and Rice (1975) is described and subsequently used for comparison with the experimental data obtained in the novel loading path tests. The use of this angle description is then extended to predicting the variation of θ with σ2 for constant σ3 and compared with experimental results in the common loading path tests (constant but different σ2 and σ3). This comparison is based on the hypothesis that angle’s variation with σ2 for a constant σ3 is due to change in the applied Lode angle (Θ) and the mean stress (σoct,f). The failure-plane angle θ, based on the localization theory by Rudnicki and Rice (1975), and rearranged by Rudnicki and Olsson (1998), takes the form: π 1 1 arcsinα 4 2    pffiffiffi 2=3 ð1 1 ν Þðβ 1 μÞ 2 N= 3 ð1 2 2ν Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where α 5 4 2 N2

θ5

(13.3)

(13.4)

(β 1 μ) is valid between 2 O3 and 1 O3. When α $ 1, θ 5 90
, corresponding to the failure mode of axial splitting or dilation bands; when α # 21, θ 5 0
, corresponding to failure in the form of pure compaction bands. The parameter N is given by pffiffiffi 2ðσ2 2 σoct Þ pffiffiffi N5 3τ oct

(13.5)

N represents the normalized intermediate principal deviatoric stress s2 (5σ2 2 σoct) by octahedral shear stress τ oct, and varies from 21 at axisymmetric extension (σ2 5 σ1) to 11 at axisymmetric compression (σ2 5 σ3). N, as a parameter that quantifies the deviatoric stress state, could be related to the third principal stress invariant, Lode angle Θ, or the stress ratio b applied in the novel tests, respectively (see Table 13.1 for the corresponding values of b and Θ to each N): N 5 2sinΘ

(13.6)

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1 2 2b NðbÞ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 b 1 b2

(13.7)

In Eq. (13.4), ν is the Poisson’s ratio; μ (5dτ oct,f/dσoct,f) is the friction coefficient, and β is the dilatancy factor. In compactive, porous rocks, for low-to-moderate σoct,f, μ . 0; for higher σoct,f, μ , 0. β is positive when the inelastic volume deformation is dilatant and negative when compactive. Rudnicki and Rice (1975) adopted a twoinvariant expression for τ oct,f 5 f (σoct,f). Hence, the failure dependence on the third stress invariant, Θ, was not considered. The two-invariant model prediction for the failure-plane angle θ is based on (β 1 μ) for given deviatoric stress states N (Eq. (13.4)): pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 3 π N 2 4 2 N  sin 2θ 2 1 pffiffiffi ð1 2 2ν Þ β1μ5 2ð 1 1 ν Þ 2 3

(13.8)

The friction coefficient and the dilatancy factor in Eq. (13.4) were not directly measurable in the experiments, but the dependence of (β 1 μ) on σoct,f could be inferred from a discretionary novel data subset (of a constant deviatoric stress state parameter N), by using the observed variation of θ with σoct,f for any constant N. Then a plausible variation of (β 1 μ) with σoct,f was obtained, which was used to compute the theoretically predicted failure-plane angle (θT, superscript “T” denoting theoretical prediction) at all constant deviatoric stress states (N). Though this variation of (β 1 μ) with σoct,f was strictly valid only for the particular N used, it was assumed that it was applicable to other subsets of N values, since the two-invariant model implies no variation of failure and plastic potential and consequently (β 1 μ) with the third stress invariant (Lode angle Θ, or interchangeably N here for the two-invariant model). In the angle prediction using the two-invariant model, the deviatoric pure shear data series (N 5 0, or b 5 1/2) was used to infer the dependence of (β 1 μ) on σoct,f. The prediction of the failure-plane angle was first applied to the novel loading path test conditions. It resulted in a continuous variation of the predicted θT with σoct,f by substituting the same continuous (β 1 μ) [ 5 f (σoct,f)] into Eq. (13.4) for each constant deviatoric stress state (N). This was used mainly for comparison of predictions between different constant deviatoric stress states. Then the prediction was compared with the novel experimental data at each constant deviatoric stress state. The failure-plane angle θ depending on σoct,f and N was further extended to obtaining angle predictions in tests using the common loading path. The experimentally obtained stress state at failure (σ1,peak, σ2, σ3) was used to calculate σoct,f and N. σoct,f then led to the value of (β 1 μ), which along with N was substituted into Eq. (13.4) to compute θT. θT and the experimentally obtained θ were compared for the same given constant σ2 and σ3. To predict the variation of θT with σ2 for constant σ3, a continuous σ1,peak vs σ2 function was required to extrapolate the failure stress condition (σ1,peak, σ2, σ3)

True triaxial failure stress and failure plane of two porous sandstones

297

where experiments were not conducted. The failure description determined by the novel loading path tests could be of use. However, any potential systematic discrepancy of the stress state at failure between experiments and theoretical descriptions may enter the failure-plane angle prediction. To that end, the empirically fitted variations of σ1,peak with σ2 for constant σ3 were used to predict a continuous θT vs σ2 trend for constant σ3.

13.6.1 Coconino sandstone For Coconino sandstone, the average Poisson’s ratio ν 5 0.25 (Ma, 2014) was used in prediction (Eq. (13.4)). However, it was suspected that Poisson’s ratio was not constant and tended to change as failure took place, and this variation might affect the value of the inferred (β 1 μ) and the resulting predictions. Hence, the influence of Poisson’s ratio was examined by inferring the dependence of (β 1 μ) on σoct,f for ν 5 0.15, 0.25, and 0.35 via Eq. (13.4) and using the data series of the deviatoric pure shear (N 5 0). Shown in Fig. 13.10, for each ν, the inferred variation of (β 1 μ) with σoct,f was generally similar. The inferred values of (β 1 μ) slightly varied with ν for a constant σoct,f, and the discrepancy between each constant ν series quickly diminished as σoct,f was increased. For a constant σoct,f 5 50 MPa, (β 1 μ) decreased by B0.5 as ν was varied from 0.15 to 0.35. All three (β 1 μ) vs σoct,f curves appeared to converge at σoct,f  460 MPa, and to diverge for larger σoct,f though it was beyond the σoct,f applied in the experiments. As illustrated

Figure 13.10 Variation of the inferred β 1 μ with σoct,f in Coconino sandstone under deviatoric pure shear (N 5 0, i.e., b 5 Δσ2/Δσ1 5 1/2) for Poisson’s ratio ν 5 0.15, 0.25, and 0.35, respectively.

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Figure 13.11 Variation of the inferred β 1 μ with σoct,f in Coconino sandstone when Poisson’s ratio ν 5 0.25, for all constant Lode angles.

by Fig. 13.10, a moderate change of ν could only contribute to a small influence on the variation of (β 1 μ). The (β 1 μ) dependence on σoct,f used in the prediction was inferred from the novel test data series conducted for deviatoric pure shear (N 5 0), under the assumption of no (β 1 μ) variation with N. However, for a constant Poisson’s ratio (50.25), such variation did exist and was shown in Fig. 13.11 for the inferred (β 1 μ) dependence on σoct,f obtained from each constant N. It showed that the discrepancy in (β 1 μ) between each N was significant. Generally, for a constant σoct,f, the inferred (β 1 μ) value decreased as N varied from 21 (axisymmetric extension) to 11 (axisymmetric compression). As σoct,f was raised, the discrepancy widened.

13.6.1.1 Novel loading path (constant σ3 and Θ) As shown in Fig. 13.12, the predicted failure-plane angle θT matched the experimental data reasonably well except for the two limiting cases: axisymmetric compression and extension (σ2 5 σ3 and σ2 5 σ1). At σ2 5 σ3, the model predicted an average of 8
higher angle throughout the applied σoct,f range (between 0 and 300 MPa). As for σ2 5 σ1, predicted angle θT was 90
for σoct,f , 200 MPa, which overestimated the measured shear band angle by at least 10
. As σoct,f increased, the predicted angle variation with σoct,f was different from that observed in the experiments. For σoct,f $ 200 MPa, the predicted angle quickly dropped from 90
to about 50
at σoct,f  465 MPa; while the measured angle declined nearly linearly throughout from about 80
to a minimum of 56
at the highest σoct,f applied (5463.8 MPa).

True triaxial failure stress and failure plane of two porous sandstones

299

Figure 13.12 Failure-plane angle (θ) variation with σoct,f for each constant Lode angle Θ in Coconino sandstone: the two-invariant model predictions (dashed curve) and the measured angles (solid dots).

13.6.1.2 Common loading path (constant σ2 and σ3) The angle prediction for Coconino common tests was based on each experimentally obtained stress condition at failure (σ1,peak, σ2, σ3). Since the angle prediction was based on the stress state at failure (σ1,peak, σ2, σ3), single abnormal σ1,peak value was likely to exacerbate the point-by-point discrepancy for a given constant σ2 and σ3. Therefore, the continuous variation of θT with σ2 was predicted using the empirically fitted σ1,peak vs σ2 trend for each constant σ3. The predicted angles were compared with the test data for all σ3 levels, as shown in Fig. 13.13. For a constant σ3, the θT vs σ2 trend was always concave upward: failure-plane angle first dropped as σ2 rose but then climbed up monotonically until σ2 5 σ1. It appeared that as σ3 was raised,

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Figure 13.13 Failure-plane angle (θ) variation with σ2 for all constant σ3 levels in Coconino sandstone tested under the common loading path: the two-invariant model predictions (dashed curves) and the measured angles (data points).

the stress ratio b where the θT reached the minimum gradually rose, and θT at σ2 5 σ1 gradually became less than at σ2 5 σ3 for the same σ3. The extreme case at σ3 5 150 MPa was that θT monotonically declined as σ2 was raised. The theoretical prediction did not agree with the experimental data. As indicated by Eq. (13.4), θT was dependent on σoct,f and N. Therefore, the predicted variation of θT with σ2, was a representation of the superimposed variation of angle with σoct,f and N, respectively. For a fixed σ3, as σ2 rose from its lower limit (σ2 5 σ3) to its upper limit (σ2 5 σ1), N decreased from 11 to 21 (causing θT to increase) and σoct,f increased (causing θT to decrease). The combined effects of σoct,f and N contributed to the concave trend (Fig. 13.13). The varying θT vs σ2 trend with increasing σ3 suggested that the effect of σoct,f gradually outweighed that of N. Despite the poor agreement at each constant σ3 level, the general variation of θT vs σ2 trend with increasing σ3 was similar to the experimental data.

13.6.2 Bentheim sandstone Based on 10 uniaxial compression tests by Ma (2014), the average Poisson’s ratio ν 5 0.36 of Bentheim sandstone was used via Eq. (13.4). The influence of varying Poisson’s ratio as failure was approached was examined by inferring the dependence of (β 1 μ) on σoct,f for ν 5 0.16, 0.26, and 0.36 using Eq. (13.4) and the data

True triaxial failure stress and failure plane of two porous sandstones

301

Figure 13.14 Variation of the inferred β 1 μ with σoct,f in Bentheim sandstone under deviatoric pure shear (N 5 0, i.e., b 5 Δσ2/Δσ1 5 1/2) for Poisson’s ratio ν 5 0.16, 0.26, and 0.36, respectively.

series of the deviatoric pure shear (N 5 0). Shown in Fig. 13.14, the inferred variation of (β 1 μ) with σoct,f for each ν was similar. As ν varied from 0.16 to 0.36, the values of (β 1 μ) declined by about 0.5 when σoct,f was between 0 and 100 MPa. The discrepancy between (β 1 μ) at different ν diminished as σoct,f was increased and appeared to approach 0 at σoct,f  200 MPa; for larger σoct,f, the discrepancy increased but not significantly. Fig. 13.14 demonstrates that a small variation of ν has limited influence on (β 1 μ) and the predicted angle. For a constant Poisson’s ratio (50.36), the effect of N on the variation of (β 1 μ) was examined in Fig. 13.15, showing the inferred (β 1 μ) dependence on σoct,f obtained for each constant N. It appeared that for a constant σoct,f (,50 MPa), the (β 1 μ) value for each deviatoric stress state remained nearly unchanged; for a constant σoct,f . 50 MPa, as (β 1 μ) dropped at an increasing rate, the difference in (β 1 μ) between each deviatoric stress states was enlarged. Generally, for a constant σoct,f 5 140 MPa, the inferred (β 1 μ) value decreased by B1.6 as the deviatoric stress state parameter N varied from 21 (axisymmetric extension) to 11 (axisymmetric compression). This discrepancy would significantly affect the prediction since the (β 1 μ) dependence on σoct,f used in the prediction was only inferred from the novel test data at deviatoric pure shear (N 5 0) under the assumption of no (β 1 μ) variation with N.

13.6.2.1 Novel loading path (constant σ3 and Θ) The predicted failure-plane angle θT in the novel tests is shown in Fig. 13.16 for each constant N. θT agreed with the experimental results reasonably well for data

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Figure 13.15 Variation of β 1 μ with σoct,f in Bentheim sandstone when Poisson’s ratio ν 5 0.36, for all constant Lode angle.

series of b 5 1/3, 1/2, and 3/4. Similar to that in Coconino sandstone, the prediction at the two extreme conditions (axisymmetric compression σ2 5 σ3, and extension σ2 5 σ1) did not match the magnitude obtained experimentally. At σ2 5 σ3, θT consistently declined with increasing σoct,f, similar to the measured trend but was off by at least 110
. The fitting was little improved for test series of b 5 1/6. For both b 5 0 (σ2 5 σ3) and b 5 1/6, the discrepancy enlarged as σoct,f increases. At σ2 5 σ1, the model predicted a 90
angle (suggesting axial splitting) for σoct,f , B100 MPa, and then a steady decline to 0
(developing compaction band) for σoct,f between about 100 and 270 MPa. The predicted values were equivalent to the measured magnitudes in the experiments, but the variation with σoct,f was significantly different.

13.6.2.2 Common loading path (constant σ2 and σ3) Same routine as used in Coconino sandstone was followed to predict failure-plane angle in the common loading path tests in Bentheim sandstone. To better illustrate the predicted effect of σ2 on failure-plane angle, a continuous variation of θT with σ2 for each constant σ3 was modeled and shown in Fig. 13.17. This continuous θT variation was based on the continuous empirically fitted experimental σ1,peak vs σ2 trend for constant σ3. For test series of σ3 , 30 MPa, θT typically first declined to a minimum then rose with increasing σ2. When σ3 . 30 MPa, all predicted θT vs σ2 trends monotonically went down as σ2 was increased.

True triaxial failure stress and failure plane of two porous sandstones

303

Figure 13.16 Failure-plane angle (θ) variation with σoct,f for each constant Lode angle Θ in Bentheim sandstone: the two-invariant model predictions (dashed curve) and the measured angles (solid dots).

13.6.3 Discussion It appeared that the two-invariant model barely represent the experimentally observed failure-plane angle variation with σoct,f in the two sandstones when the deviatoric stress state parameter N approached its two extremes: 21 (σ2 5 σ1) and 11 (σ2 5 σ3). When N approached 11 (σ2 5 σ3), the prediction tended to overpredict the angle, though generating a similar variation of θT with σoct,f; when N approached 21 (σ2 5 σ1), the variation of θT with σoct,f was different in that the model over-predicted θT at low σoct,f and underestimated θT at high σoct,f. The reason for the discrepancies at σ2 5 σ3 and σ2 5 σ1 was unclear. This discrepancy was suspected to be directly responsible for the mismatch of θ vs σ2 trend between the predicted and the observed in the common tests. As the prediction process only

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Figure 13.17 Failure-plane angle (θ) variation with σ2 for all constant σ3 levels in Bentheim sandstone tested under the common loading path: the two-invariant model predictions (dashed curves) and the measured angles (data points).

required the state of stress at failure regardless of the applied loading path, the stress condition at failure (σ1,peak, σ2, σ3) in the common tests first led to (σoct,f, N), and the latter served as the direct input to the angle prediction via Eq. (13.4). In this sense, the variation of θT with σ2 was largely affected by the predicted variation of θT with σoct,f and N. Therefore, the over-prediction and underestimation as N approached 11 and 21, respectively, induced a typical descending-then-ascending θT vs σ2 trend for each constant σ3 series, which differed significantly from the experimental observations. It seemed that the discrepancy was mainly due to the excessive variation of the predicted θT with N for a constant σoct,f. And it was directly related to the variation of (β 1 μ) with σoct,f. The poor predictions suggested a discrepancy between the actual (β 1 μ) value and the inferred. It should be noted that the inferred (β 1 μ) variation with σoct,f used in the prediction was based on the novel test data series of deviatoric pure shear (N 5 0), under the assumption of no (β 1 μ) variation with N. However, it was shown that such variation did exist in both rocks (Figs. 13.11 and 13.15), and it seemed that the discrepancy in (β 1 μ) between each N was significant. The use of (β 1 μ) values under N 5 0 in predictions may serve as a reasonable average but would inevitably worsen the prediction as N approaches its lower and upper boundaries. It revealed the limitation of the two-invariant model, that is, assuming no variation of the (β 1 μ) values with the deviatoric stress state (N).

True triaxial failure stress and failure plane of two porous sandstones

305

In view of this, a model that allows the variation of (β 1 μ) with N, that is, the deviatoric stress state (or Lode angle Θ), might improve the predictions. Another possibility is that predictions of this theory for smooth failure surface models are known to have deficiencies for axisymmetric stress states (Rudnicki and Rice, 1975).

13.7

Concluding remarks

We have conducted true triaxial experiments in two porous sandstones aimed at examining failure under a general compressive stress field. The common loading path series of tests in which σ2 and σ3 were held constant confirmed the results previously obtained in less porous clastic rocks and crystalline rocks. These results demonstrate that failure in porous sandstones is dependent not only on σ3, but also on σ2. Common to both sandstones under true triaxial loading, σ1,peak first ascends then descends with σ2, and failure-plane angle increases consistently with σ2. The novel loading path series of tests in which the Lode angle was kept constant revealed the respective effects of Lode angle and mean stress to failure stress and failure-plane angle. We found in both sandstones that given constant octahedral shear stress the mean stress required for failure typically increased as Θ varied from 130
to 230
. The failure-plane angle monotonically decreases with mean stress and also decreases with Lode angle. Differences exist between the two sandstones in that Bentheim sandstone is more compactive and experienced a more obvious brittleductile transition. Bentheim sandstone entered early ductile regime through the development of lowangle shear-enhanced compaction bands (θ , 40
) under high σ3, while the early ductile regime was never reached in Coconino sandstone for the maximum σ3 applied. We employed the bifurcation theory by Rudnicki and Rice (1975) to model the failure-plane angle variation. The theoretical prediction for novel loading path tests is promising by faithfully replicating the failure-plane angle variation with mean stress and Lode angle. The model was then extended to predict the failure-plane angle variation with σ2 for constant σ3. The prediction also qualitatively agrees with their experimental trend. This comparison suggests the variation of failureplane angle with σ2 could be fundamentally considered as a result of angle variation with both Lode angle and mean stress.

Acknowledgments This research was supported by the National Science Foundation Grant EAR-0940323 (for BCH and XM) and Grant EAR-0940981 (for JWR).

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Index

Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively. A Acid fracturing, coupling in, 58 Anisotropic poroplasticity, 25 anisotropic plastic behavior of rocks, 35 41 effects of temperature on anisotropic rocks, 41 45 general framework of poroplastic modeling, 28 31 effective stress concept in poroplasticity, 30 31 typical porous rock, experimental investigation on, 32 34 Axial and radial strains, 43f

B Barnett Shale, 6 7 Bedding planes and parallel fractures, 108 110 Bentheim sandstone, 287, 291 292, 300 302 common loading path, 302 novel loading path, 301 302 Bifurcation theory, failure-plane angle via, 295 305 Bentheim sandstone, 300 302 common loading path, 302 novel loading path, 301 302 Coconino sandstone, 297 300 common loading path, 299 300 novel loading path, 298 Bingham fluid, 55 56 Biot coefficients tensor, 204 205 Biot poroelastic theory, 124 Biot’s coefficient tensor, 31 Biot’s coefficients, 205 second-order tensor of, 29 Biot’s modulus, 29 Biot’s theory of consolidation, 204 205

Bonded-particle model (BPM), 147 148, 272 273 Boundary element method (BEM), 105, 198 200, 259, 270 271, 273 276 Broyden Fletcher Goldfarb Shanno (BFGS) methods, 184 185

C Carman Kozeny equation, 81 Carter’s law, 51 53 Cauchy stress tensor, 36, 201 Chemical dissolution process, simulation of, 83 84 Clausius Duhem inequality, 201 Closed cracks, analysis of, 7 8 CO2 brine rock reaction, indentation tests on samples after, 74 76, 74f Coal hydraulic fracturing, 97 98 laboratory tests, 100f Coconino sandstone, 287, 291 292, 297 300, 302 common loading path, 299 300 novel loading path, 298 Cohesive traction separation laws, 119 120 Cohesive zone finite-element method (CZ-FEM), 106 107 of hydraulic fracture natural fracture interaction, 108 of 3-D T-shaped hydraulic fracture, 107 Cohesive zone models (CZMs), 106, 117, 198 200, 233 234, 268, 276 277 hydraulic fracturing, 121 128 cohesive zone model as fracture propagation criterion, 126 128 fluid flow, 123 124 rock deformation, 124 125 model geometry and input data, 129 131 results and discussion, 131 140

310

Cohesive zone models (CZMs) (Continued) influence of cohesive loading modulus on process zone, 132 135 influence of confining stresses on process zone, 135 137 influence of pore pressure and diffusion on process zone, 137 138 influence of softening modulus on process zone, 139 140 Compactive failure, 287 Computer-aided design (CAD) model, 207 208 Contact bonds, 147 148 Continuum damage mechanics (CDM), 197, 233 of porous rocks, 203 207 for simulation results, 207 209 Coupled fluid flow in porous rocks, 257 fractured rocks, numerical modeling of, 266 278 literature review of different approaches for, 266 276 XFEM approach, 276 278 numerical modeling of continuous media applied to rocks, 258 266 numerical methods for coupled fluid flow modeling, 258 259 numerical scheme, 262 264 performances, stability, and accuracy, 264 266 in rocks with finite element modeling, 259 262 prospects, 278 Coupling in hydraulic fracturing simulation, 47 acid fracturing, coupling in, 58 coupling fluid and solid, 53 54 coupling in reservoir geomechanics, 47 50 coupling proppant transport and placement, 55 57 fluid-driven fracture propagation in rocks, 47 fracture-matrix fluid exchange, 50 53 thermal coupling, 57 Crack closure stress, 198 Crack geometry and in-situ stresses, 8f Crack initiation and propagation, 119, 121, 270

Index

Creep plastic strain tensor, 76 77 Creep tests with injection of CO2 alone, 72, 73f, 88 89 Creep tests with injection of CO2 brine, 73 74, 73f, 88 89

D Damage tensor, 241 242 Darcy’s law, 51 52, 56, 78, 81, 276 Decoupled models, 49 Dirac-delta distribution, 179 180 Discrete element method (DEM), 145, 272 273 features of, 145 146 for fracture networks, 159 173 discrete fracture network model, 159 160 hydromechanical calculation, 160 162 particle tracking method for solute transport, 162 165 stress effects on fluid flow and solute transport, 165 171 particle-based DEM for rock fractures and matrix, 147 159 basic features of, 147 148 single fracture damage during shear, 148 154 thermal-induced rock weakening and failure, 154 159 Discrete fracture network (DFN), 272 273 Displacement discontinuity method (DDM), 10 11, 105, 270 Double porous rock macroscopic criterion of, 217 224 homogenization from mesoscale to macroscale, 219 224 homogenization from microscopic to mesoscopic scale, 218 219 nonassociated elastoplastic model for, 224 227 Drucker Prager criterion, 218, 223 Dynamic continuum damage mechanics, 233 234 Dynamic crack growth models, 233 234 Dynamic fracture mechanics, 233 constitutive relations for microcracks and microvoids, 244 246

Index

continuum mechanics context, dynamic fracture prediction techniques in, 240 242 equation of state (EOS), 237 238 failure modes associated with dynamic problems, 243 244 kinematics of elastic and inelastic deformations in porous rocks, 235 237 low-to-high strain rate constitutive modeling, 238 239 mean stress effect on the strength of rock and mechanical properties, 239 240 numerical implementation, results, and discussions, 246 250 techniques in continuum mechanics context, 240 242

E Effective stress, 28, 34, 124 in poroplasticity, 30 31 Terzaghi’s effective stress, 30 32 Effective stress tensor, 30 31, 204 205 Elastic and inelastic deformations in porous rocks, kinematics of, 235 237 Elastic behavior, 27 29 Elastic deformation, 76 77 Elastic strain rate, 236 Elasticity theory, 122 Elastic softening model, 134 135 Elasto-plastic rock behavior, 207 208 Element free Galerkin (EFG) method, 268 269 Embedded finite element method (E-FEM), 177 185 finite element (FE) discretization, 183 185 Hu-Washizu formulation, 181 183 kinematically optimal symmetric (KOS) formulation, 183 statically and kinematically optimal nonsymmetric (SKON) formulation, 183 statically optimal symmetric (SOS) formulation, 182 183 induced fracture networks, application to, 185 187 model description, 185 187

311

kinematics description of strong discontinuities, 178 181 3D multicracking, application to, 188 194 Enhanced assumed strain method (EAS), 180 181 Equation of state (EOS), 237 238 Equivalent continuum models, 146 Extended finite element method (X-FEM), 105 106, 177 178, 198 200, 267, 273 276

F Failed elements, 53 Failure characteristics, 287 Failure-plane angle, 286, 286f, 293 295 via bifurcation theory. See Bifurcation theory, failure-plane angle via Fick law, 262 Field-scale mineback experiments in coal, 100 Finite difference method (FDM), 49, 129, 259, 265 Finite element (FE) discretization, 183 185 Finite element finite volume method (FEFVM), 270 Finite element method (FEM), 105, 177, 258, 264 Finite volume method (FVM), 49, 258 259 Finite-element analysis (FEA) model, 140 141, 198 200, 235, 246 Finite-element cohesive zone model, 106 Flow wetted surface (FWS), 164 Fluid and solid, coupling, 53 54 Fluid flow, 17, 103 104, 122 124, 161 162, 207 208 Fluid flow and solute transport, stress effects on, 165 171 advection, 165 167 effects of roughness on macroscopic flow and transport, 171 173 flow results, 165 hydrodynamic dispersion, 167 170 matrix diffusion, 170 171 Fluid-driven fracture propagation in rocks, 47 Fluid rock interactions, 63 64 Force penetration curves, 75 76, 75f

312

Fourth rank tensor, 29 Fourth-order tensor, 29, 77 Fracture aperture, 47, 50, 56 Fracture mechanics criterion, 245 246 Fracture networks, discrete element method for, 145, 159 173 discrete fracture network model, 159 160 hydromechanical calculation, 160 162 particle tracking method for solute transport, 162 165 advection, 163 hydrodynamic dispersion, 163 matrix diffusion, 164 165 stress effects on fluid flow and solute transport, 165 171 advection, 165 167 effects of roughness on macroscopic flow and transport, 171 173 flow results, 165 hydrodynamic dispersion, 167 170 matrix diffusion, 170 171 Fracture process zone (FPZ), 106 Fracture propagation, 10 11, 18, 19f, 119, 122 123, 271 272 Fractured rocks, numerical modeling of, 266 278 literature review of different approaches for, 266 276 boundary elements method, 273 276 discrete element method, 272 273 extended finite element method, 273 276 hybrid methods, 269 270 interface element methods, 267 268 meshless methods, 268 269 XFEM approach, 276 278 numerical implementation and numerical scheme, 277 278 physical model, 276 Fracture-matrix fluid exchange, 50 53 Fracture-parallel normal stress distributions, 102 103 Fully coupled model, 49, 140 141

G Galerkin least-squares methods, 265 266 Galerkin’s finite-element approach, 10 11 Generalized Darcy law, 260 262

Index

Geo-engineering applications, 63 64 Geometrical domain and boundary conditions, 38, 39f

H Heaviside function, 178 179, 272 273, 277 278 Helmholtz-free energy function, 201 202 Homogenization from mesoscale to macroscale, 219 224 from microscopic to mesoscopic scale, 218 219 Homogenized stiffness tensor, fourth-order, 204 205 Hook’s law, 236, 265 Hopkinson-Bar method, 234 Horizontal stress, effective, 38 40 Horizontal wells, reservoir stimulation by hydraulic fracturing of, 3 6 Hu-Washizu formulation, 181 183 kinematically optimal symmetric (KOS) formulation, 183 statically and kinematically optimal nonsymmetric (SKON) formulation, 183 statically optimal symmetric (SOS) formulation, 182 183 Hybrid methods, 269 270 combination of numerical methods, 270 partition of unity finite element method, 269 270 Hydraulic fracture (HF) growth in naturally fractured (NF) rock, 93 analysis of HF NF interaction, 101 105 fracture interaction criteria, 101 105 future directions, 111 112 interaction and crossing, 95 96 natural fractures and hydraulic fractures, 93 95 numerical modeling of HF NF interaction, 105 111 design modeling, 110 111 extended finite-element model, 106 finite-element cohesive zone model, 106 MineHF2D model, 105 106 observed fracture geometries, model tests for, 107 110

Index

observations and experiments, 96 101 laboratory experiments, 99 101 mineback studies, 97 99 monitoring results, 99 Hydraulic fracture spacing optimization using numerical simulations, 10 15 Hydraulic fracturing (HF), 121 128, 197 cohesive zone model as fracture propagation criterion, 126 128 conceptual models, 6 9 rock failure and the stimulated volume, 7 9 continuum damage mechanics (CDM) of porous rocks, 203 207 drained and/or undrained conditions, 200 fluid flow, 123 124 influence of natural fractures on, 16 17 micro-seismic monitoring of, 15 reservoir stimulation by, of horizontal wells, 3 6 rock deformation, 124 125 simulation results, CDM model for, 207 209 simulation techniques and, 198 200 thermodynamic principles, 200 207 Hydraulic fracturing simulation coupling in. See Coupling in hydraulic fracturing simulation Hydraulic fracturing simulator coupling of thermal, hydraulic, and mechanical processes in, 59f Hydromechanical calculation, 160 162 Hydro-mechanical coupling (HM-coupling), 47 48 in coupled geomechanical simulation, 48f HM-coupled reservoir models, 47 48 in reservoir geomechanics, 50f Hydromechanical chemical (HMC) coupling behavior, 63 64, 76 88 general framework, 76 78 numerical application, 83 88 chemical dissolution process, simulation of, 83 84 hydro-mechanical chemical coupling behavior, simulation of, 86 88 mechanical behavior, simulation of, 84 86 of sandstone, 69 76

313

creep tests with injection of CO2 alone, 72, 73f creep tests with injection of CO2 brine, 73 74, 73f indentation tests on samples after CO2 brine rock reaction, 74 76, 74f sandstone, special model for, 79 83 mass-transfer modeling, 81 82 mechanical modeling, 79 81 poromechanical modeling, 83 porosity evolution and chemical damage, 82 83 simulation of, 86 88 “Hydro-shear” conceptual model, 7 Hydrostatic and triaxial compression tests, 41 42 Hydrostatic poromechanical tests, 32f I In-situ stresses, 8f, 145 Interface element methods, 267 268 Interface elements (IEs), 128, 267 Intrinsic dissipation, 29 positiveness of, 31 Iteratively coupled model, 49 J Johnson Cook (J C) model, 238 239 K Kinematically enhanced strain (KES), 179 180 Kinematically optimal symmetric (KOS) formulation, 183 Kinematics description of strong discontinuities, 178 181 L L2-orthogonal condition, 180 181 Lac du Bonnet granite, 148f, 156 Lagrangian functional, 202 Leakoff, 50 53 Limit analysis method, principle of, 216 217 Linear elastic fracture mechanics (LEFM), 106, 119 120, 198 200, 270

314

Lixhe chalk, 227 230 Load vector, normalized, 36 Loading unloading cycle, 33f, 81 Lode angle, 286, 289 290, 293, 295, 305 Low-to-high strain rate constitutive modeling, 238 239 Lubrication theory approximation, 53 54, 56

M Macauley bracket, 80 Mass-transfer modeling, 81 82 Mechanical aperture, 53 54, 171 Mechanical behavior, simulation of, 84 86 Meshless methods, 268 269 Mesoporosity, 225, 230 231 Microcrack and microvoids, constitutive relations for, 244 246 Micro macro model, experimental validation of, 229 231 MICROPE, 75 Microporosity, 223 225, 230 231 Micro-seismic monitoring of hydraulic fracturing, 15 Mie-Gru¨neisen EOS, 237 MineHF2D model, 102 106 “Modified zipper” fracturing (MZF), 4 Mohr Coulomb criterion, 30 31, 180 181, 184 187, 189 model description, 185 187 MPG (material property group), 164, 170 171 Multiple hydraulic fractures, mechanical interactions of, 9 20 hydraulic fracture spacing optimization using numerical simulations, 10 15 natural fractures, influence of on hydraulic fractures, 16 17 rock anisotropy, role of, 15 3D effects, importance of, 18 20 Multiscale modeling approaches and micromechanics of porous rocks double porous rock, macroscopic criterion of, 217 224 homogenization from mesoscale to macroscale, 219 224 homogenization from microscopic to mesoscopic scale, 218 219

Index

limit analysis method, principle of, 216 217 nonassociated elastoplastic model for double porous material, 224 227 typical porous rock, application to, 227 231 micro macro model, experimental validation of, 229 231 model’s parameters, identification of, 228 229 N Natural fractures and hydraulic fractures, 16 17, 93 95 Net pressures, 122, 133, 135, 138, 138f Neumann surface, 181 182 Newtonian fluid, 55 56, 123, 271 272 Newton Raphson algorithm, 266 Niobrara chalk formation, 11t, 18 Normalized load vector, 36 Numerical manifold method (NMM), 269 O One-dimensional (1D) fracture, 50, 51f One-way coupling, 48 49 OpenT criterion, 103 104 P Parallel bonds, 147 148, 147f Partial differential equations (PDEs), 258, 262 Partially coupled model, 49 Particle tracking method for solute transport, 162 165 advection, 163 hydrodynamic dispersion, 163 matrix diffusion, 164 165 Particle-based DEM for rock fractures and matrix, 147 159 features of, 147 148 single fracture damage during shear, 148 154 fracture segments with gouge, 151 154 fracture segments without gouge, 149 150 thermal-induced rock weakening and failure, 154 159

Index

Pe´clet number, 78, 81 Plastic deformation, 27, 38 40, 76 77 and failure process, 37 Plastic effective stress coefficient, 32 Plastic flow rule, 29 30 Plastic hardening law, 38, 227 228 Plastic strains, 32 33 Point interpolation method (PIM), 268 269 Poisson’s ratio, 75, 205 207, 225, 297 298, 300 301 Poly methyl methacrylate (PMMA), 101 Polycrystalline diamond compact (PDC), 247 250 Poroelasticity, 200 theory of, 124 125 Poromechanical modeling, 83 Poroplastic coupling, 38 Poroplastic loading—unloading cycle, 33f Poroplastic modeling, general framework of, 28 31 effective stress concept in poroplasticity, 30 31 Poroplasticity anisotropic. See Anisotropic poroplasticity pressure-sensitive, 205 Porosity evolution and chemical damage, 82 83 Porosity ratio, 225f Porous rocks, 227 231 experimental investigation on, 32 34 micro macro model, experimental validation of, 229 231 model’s parameters, identification of, 228 229 Porous sandstones, 285 Pressure permeability fluid loss positive feedback cycle, 93 94 Pressure-sensitive Mohr Coulomb type criterion, 30 31 Principal stress invariants, true triaxial failure stress in terms of, 293 Principal stresses, true triaxial failure stress in terms of, 290 291 Process zone, 119 120 influence of cohesive loading modulus on, 132 135 influence of confining stresses on, 135 137

315

influence of pore pressure and diffusion on, 137 138 influence of softening modulus on, 139 140 Proportionality factor, 244 Proppant, 121 Proppant transport and placement, coupling, 55 57 Pulse test method, 64 67 Q Quadratic interpolations, 129 R Reaction rate law, 82 Renshaw and Pollard criterion, 103 Representative elementary volume (REV), 146 Representative volume element (RVE), 216 218, 217f Reservoir geomechanics, coupling in, 47 50 Reservoir stimulation by hydraulic fracturing of horizontal wells, 3 6 Reservoir sample reservoir system, 67 69 Reynolds number, 171 Rigid softening model, 134 135 Rock anisotropy, 3, 15 Rock deformation, 124 125 Rock failure and the stimulated volume, 7 9 Rock fracture, 148, 150f, 233 Rock materials, 63 Rocks, anisotropic plastic behavior of, 35 41 S Sandstone, special model for, 79 83 mass-transfer modeling, 81 82 mechanical modeling, 79 81 poromechanical modeling, 83 porosity evolution and chemical damage, 82 83 Scalar anisotropy parameter, 35 37 Secondary cracks, formation of, 7 8 Shear slip, 3, 8

316

Shear-induced dilation of fractures, 93 94, 99 Slip plane models, 248 250 Small-scale plasticity, 120 121 Smooth joint model (SJM), 272 273 Spurt loss, 53 Statically and kinematically optimal nonsymmetric (SKON) formulation, 181, 183 Statically optimal symmetric (SOS) formulation, 182 183 Steady permeability tests, 65 66, 65f Steady-state flow method, 64 65 Stimulated volume, 4, 7 9 Strain field, 179 181 Streamline upwind Petrov Galerkin (SUPG), 265 266 Stress based criterion, 245 246 Stress equivalence principle, 27, 30 31 Stress interference between cavities and fractures in rock, 9 10 Stress ratio, defined, 165 Stress shadow, 4 6, 10 12 Stress flow transport simulation, 165 Stress-induced permeability evolutions and erosion damage of porous rocks, 63 hydromechanical chemical coupling behavior, numerical simulations of, 76 88 chemical dissolution process, simulation of, 83 84 general framework, 76 78 hydro-mechanical chemical coupling behavior, simulation of, 86 88 mechanical behavior, simulation of, 84 86 sandstone, special model for, 79 83 laboratory tests, 64 76 hydro-mechanical chemical coupling behavior of sandstone, 69 76 steady permeability tests, 65 66, 65f transient pulse tests, 66 69, 68f Stress pressure cycles, 32 33 Stress strain curves, 34f, 42 45, 43f, 44f, 66, 67f, 70f Stress strain relations, 29 Synthetic rock mass (SRM), 272 273

Index

T Tensile “wing cracks”, 7 Tension compression asymmetry, 223 224 Terzaghi’s effective stress, 30 32 Thermal coupling, 57 Thermal hydrological mechanical chemical (THMC) processes, 145, 148 Thermal hydro-mechanical and chemical (THMC) coupling testing system, 69 70, 71f Thermal-induced rock weakening and failure, 154 159 Thermo-HM (THM)-coupled model, 57 Thermohydromechanical chemical (THMC) couplings, 265 266 Thermo-mechanical coupling, 57 3D effects, importance of, 18 20 3D hydraulic fracturing models, 52 53 3D multicracking, application to, 188 194 Tournemire shale, 35, 38, 41 42 Transient pulse tests, 66 69, 68f Triaxial compression test, 32 33, 69 Triaxial testing techniques, 234 True triaxial failure stress, 285 experiment procedures, 288 290 common loading path, 288 novel loading path, 289 post-test procedure, 289 290 testing program, 290 materials, 287 prediction of failure-plane angle via bifurcation theory, 295 305 Bentheim sandstone, 300 302 Coconino sandstone, 297 300 discussion, 303 305 under common loading path, 290 292 failure mode and failure-plane angle, 291 292 in terms of principal stresses, 290 291 under novel loading path, 293 294 failure-plane angle, 293 294 in terms of principal stress invariants, 293 Two-dimensional (2D) fracture, 51 Two-way coupling, 47 49 Typical porous rock, 28, 31 experimental investigation on, 32 34

Index

317

U Uniaxial compression strength, 38 variation of, 39f Unity finite element method, partition of, 269 270 Universal distinct element code (UDEC), 160 162

W Wavy crack models, 233 234 Wing cracks, 7 8, 9f Woodford shale, 13

V Viscoplastic loading surface, 80 Viscoplastic yield surface, 80 Von Mises flow stress, 238 239 Von Mises stress distribution, 208 209 Von-Mises criterion, 30 31

Y Yield stresses, 34f Young’s modulus, 225

X X-ray diffraction technique, 65

Z “Zipper” fracturing technique, 3 4