Compaction bands due to grain crushing in ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116, B08203, doi:10.1029/2011JB008265, 2011

Compaction bands due to grain crushing in porous rocks: A theoretical approach based on breakage mechanics Arghya Das,1 Giang D. Nguyen,1 and Itai Einav1 Received 27 January 2011; revised 26 April 2011; accepted 3 May 2011; published 6 August 2011.

[1] Grain crushing and pore collapse are the principal micromechanisms controlling the physics of compaction bands in porous rocks. Several constitutive models have been previously used to predict the formation and propagation of these bands. However, they do not account directly for the physical processes of grain crushing and pore collapse. The parameters of these previous models were mostly tuned to match the predictions of compaction localization; this was usually done without validating whether the assigned parameters agree with the full constitutive behavior of the material. In this study a micromechanics‐based constitutive model capable of tracking the evolving grain size distribution due to grain crushing is formulated and used for a theoretical analysis of compaction band formation in porous rocks. Linkage of the internal variables to grain crushing enables us to capture both the material behavior and the evolving grain size distribution. On this basis, we show that the model correctly predicts the formation and orientation of compaction bands experimentally observed in typical high‐porosity sandstones. Furthermore, the connections between the internal variables and their underlying micromechanisms allow us to illustrate the significance of the grain size distribution and pore collapse on the formation of compaction bands. Citation: Das, A., G. D. Nguyen, and I. Einav (2011), Compaction bands due to grain crushing in porous rocks: A theoretical approach based on breakage mechanics, J. Geophys. Res., 116, B08203, doi:10.1029/2011JB008265.

1. Introduction [2] Localized narrow deformation zones are widely observed in porous rocks. Mollema and Antonellini [1996] discovered the existence of localized deformation bands caused by pure compaction without being accompanied by any shear deformation within the Aeolian sandstone; they termed it “compaction band.” Afterward the formation of pure compaction bands were observed in many other investigations through laboratory experiments on various kinds of rock samples [e.g., Olsson, 1999; Haimson, 2001; Tembe et al., 2008; Wong et al., 2001]. In different experimental studies compaction bands were also seen in nongeological material, such as metal foam [Bastawros et al., 2000] and polycarbonate honeycomb [Papka and Kyriakides, 1998]. According to Rudnicki and Rice [1980], the formation of such localization bands initiates either due to stress concentration in the place of local heterogeneities or any abrupt change in strength of material in a localized zone. [3] Menéndez et al. [1996] and Wu et al. [2000] provided valuable experimental observations regarding the micromechanics driving the formation of planar deformation zones in porous rocks. Various mechanisms were identified, including: grain crushing, grain sliding, bond breaking and pore collapse. Shearing at low confining pressures facilitates 1

School of Civil Engineering, University of Sydney, Sydney, Australia.

Copyright 2011 by the American Geophysical Union. 0148‐0227/11/2011JB008265

the fracture of grain bonding cement, allowing the grains to rotate and slip, which could be followed by the flow of granulated materials. During this bond breaking process, the mobilized shear strength of the material will obviously reduce [Menéndez et al., 1996; Wong et al., 1997]. In contrast, high confining pressures frustrate grain rotation and slip when shear loading is applied. Instead, the contacting grains tend to crush under the pressure, leading to the rearrangement of fragments, which further reduces the porosity and consequently hardens the material [Menéndez et al., 1996; Wong et al., 1997]. In both loading cases (under low and high pressures), pore collapse acts as a passive mechanism facilitated by either a bond breaking or a grain crushing event. At a macroscopic level these failures can be classified as brittle failures followed by shear enhanced dilation at low confining stresses, or cataclastic flow with shear enhanced compaction in high‐pressure regimes [Wong et al., 1997]. Generally, compaction bands and shear enhanced compaction bands are observed in the transition zone between brittle and ductile behavior [Wong et al., 2001]. [4] The theoretical prediction of compaction band formation in porous rocks can be based on either continuum or discrete approaches. Discrete approaches [e.g., Katsman et al., 2005; Katsman and Aharonov, 2006; Marketos and Bolton, 2009; Wang et al., 2008] offer an excellent access to insights into the formation and propagation of compaction band at the grain scale, but they critically depend on constitutive assumptions at the microscale. Besides, they are very computationally demanding, tending to be subjected to strong numerical noise when the number of grains is not suffi-

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DAS ET AL.: COMPACTION BANDS AND GRAIN CRUSHING

ciently high, and mostly suffering from numerical integration inaccuracies due to the highly nonlinear nature of this complex multibody system. On the other hand, continuum approaches have also been widely used [Olsson, 1999; Issen and Rudnicki, 2000; Rudnicki, 2004; Challa and Issen, 2004; Chemenda, 2009], and have the capacity to provide access to closed form solutions under particular conditions, that can clearly map the extent of the various effects. Issen and Rudnicki [2000] made a successful attempt toward the prediction of compaction band formation using the bifurcation approach proposed by Rudnicki and Rice [1975] for shear localization. Challa and Issen [2004] further modified this approach and proposed a constitutive model possessing two yield surfaces, one of which aims to capture the inelastic compacting behavior during the formation of the compaction band. Grueschow and Rudnicki [2005] elaborated this aspect of the model even further by introducing an elliptic cap yield surface. Their parametric study on the aspect ratio of the ellipse showed that for better prediction of localization mode a cap is required which becomes steeper under continued loading. We wish to stress that the parameters of these continuum models may be fitted to match the onset and orientation of compaction bands, but this is only one side of the story. Without readjusting those parameters, can the same models predict experimental stress‐strain curves? This is not a trivial question, but one that we are willing to explore in the current paper. [5] According to Aydin et al. [2006], a suitable constitutive model should be able to track the evolving microstructures of porous rocks due to grain crushing, pore enlargement and pore collapse, grain boundary sliding and pressure solution. Also the model needs to be capable of capturing the experimentally observed onset of compaction localization in the transitional regime between brittle fracture and ductile faulting [Wong et al., 1997]. In addition, as specifically addressed by Wong et al. [2001], it is essential for constitutive models to have multiple distinct dissipative mechanisms in the transitional zone (between brittle fracture and ductile cataclastic flow) to faithfully describe the behavior and the formation of localized deformation bands in porous rocks. Given the above requirements, it can be seen that most of the current continuum models dedicated to this issue are not adequate, as they are based on conventional plasticity theory [Olsson, 1999; Issen and Rudnicki, 2000; Rudnicki, 2004; Challa and Issen, 2004; Chemenda, 2009], or rely on the extension of fracture mechanics (the anticrack model by Sternlof et al. [2005]). Those models cannot explain the physics dictating the porous rock behavior and compaction band formation. This is due to the lack of explicit links between the only internal variable (plastic strain) in these models and the two distinguishable underlying micromechanical processes of grain crushing and pore collapse. As a consequence, these plasticity‐based models are unable to track the evolution of the grain size distribution during grain crushing, resulting in difficulties in predicting correctly the permeability change due to such a microscopic process. [6] In this study, we approach the issue of constitutive modeling of porous rocks and the formation of compaction bands from a rather more physical angle, which distinguishes itself from elastoplasticity attempts. We use a con-

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stitutive model based on the recently developed breakage mechanics theory for crushable granular material [Einav, 2007a, 2007b]. The main feature of this theory is that it can take into account the grain crushing phenomena through the evolving grain size distribution (gsd). This gsd is micromechanically linked with the macroscopic continuum behavior through the breakage measure B, as an internal variable of the continuum model, so that the grain size distribution can be continuously tracked during the deformation and associated grain crushing process. Moreover the theory is also able to capture the effects of pore collapse on the macroscopic behavior of the material. Models based on breakage mechanics theory possesses only a few physically identifiable parameters and were successfully used to predict the mechanical response and permeability reduction of porous rocks [Nguyen and Einav, 2009]. [7] A model based on breakage mechanics theory is formulated and used in this research to study the onset and orientation of compaction bands in sandstones. We note that these grain‐crushing‐induced compaction bands can only be formed in high‐pressure regimes, where we consider the current breakage model to be adequate. Cement failure, which affects the cohesive strength, only dominates the material behavior in low‐pressure regime [Menéndez et al., 1996], and therefore is expected to have insignificant contribution to the model response and the prediction of compaction bands at high pressures. We start the main parts of the paper with a brief introduction of breakage mechanics, followed by the description of a simple breakage model. The validity of this model is assessed against experimental data. The model parameters are determined for a typical high‐porosity sandstone, and the model is used to predict compaction localization observed in this rock. The localization analysis is based on the loss of positiveness of the determinant of the strain localization tensor (or acoustic tensor) [Rudnicki and Rice, 1975]. Using different loading conditions we determine favorable conditions for compaction localization and links between the material microstructures and the formation of compaction band. Agreements between numerical predictions and published experimental observations [Baud et al., 2006; Wong et al., 2001] for high‐porosity sandstone are found. A parametric study is carried out to explore the effect of the variation of initial microstructures on the formation of compaction localization.

2. Breakage Mechanics 2.1. Fundamentals of Breakage Mechanics [8] The breakage mechanics theory for crushable granular materials [Einav, 2007a, 2007b] is built on the micromechanics of grains, using statistical homogenization for the upscaling procedure. It relies on the following four assumptions: [9] 1. The elastic strain energy stored in the grains scales with their surface areas [Einav, 2007a]. [10] 2. The crushing process gradually shifts the initial grain size distribution toward the ultimate one, which can take any shape, but is conveniently assumed to follow a fractal law. This assumption is often supported by field observations of geological fault gouges [Sammis et al., 1987]. [11] 3. The current grain size distribution (g(d, B)) is determined by linear interpolation between the initial (g0(d))

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and ultimate (gu(d)) grain size distribution (where d is the grain diameter) [Einav, 2007a]:

[15] Equations (4) and (5) can be written in tensorial form as

g ðd; BÞ ¼ ð1 BÞg0 ðd Þ þ Bgu ðd Þ

ij ¼ ð1 #BÞDijkl "kl "pkl

ð1Þ

[12] 4. The dissipation due to grain crushing is equal to the loss in the residual breakage energy [Einav, 2007a]. [13] In equation (1), B is the internal variable of the continuum breakage model, representing the degree of grain crushing. As can be seen, the current evolving gsd can be tracked at any stage of the deformation process, given the initial and ultimate ones. Assumptions 1–3 are used in a statistical homogenization scheme to upscale the grain‐scale energy potential to obtain the macro energy potential of the continuum model. The grading index # is a result of this statistical homogenization, and can be obtained from the initial and ultimate gsd’s as: # ¼ 1 J2u =J20

ð2Þ

where J20 and J2u are second‐order moments of the initial and final gsd’s [Einav, 2007a]. # admits values from 0 to 1, measuring the “distance” between the initial and ultimate gsd’s. The energy balance postulate 4 enables us to derive a yield surface, on the basis of the loss of the energy thermodynamically associated with the breakage internal variable, which reveals surprising analogies with Griffith’s well‐known theory in fracture mechanics [Einav, 2007d]. 2.2. Constitutive Models Based on Breakage Mechanics [14] A simple breakage model proposed by Einav [2007c] and later used and improved by Nguyen and Einav [2009] is adopted and further enhanced in this study. We use the following standard notations: mean effective stress p, shear stress q, total volumetric strain "v and elastic volumetric strain "ev, total shear strain "s and elastic shear strain "es . As a result of the statistical homogenization using assumptions 1–3, the Helmholtz free energy potential takes the form: Y ¼ ð1 #BÞ

1 e2 3 e2 K" þ G"s 2 v 2

ð3Þ

where sij (i, j = 1, 2, 3) is Cauchy stress tensor; "kl and "pkl are the total and plastic strain tensors, respectively; Dijkl is the linear (isotropic) elastic tangent stiffness tensor; [16] On the basis of postulate 4 in section 2.1, the following breakage criterion for isotropic loading conditions can be obtained [Einav, 2007b]: yB ¼

p¼

@Y ¼ ð1 #BÞK"ev @"ev

ð4Þ

q¼

@Y ¼ 3ð1 #BÞG"es @"es

ð5Þ

EB ¼

2 @Y # p q2 ¼ þ @B 2ð1 #BÞ2 K 3G

ð6Þ

ð1 B Þ2 E B 10 Ec

ð8Þ

where Ec is the critical breakage energy (of stress dimension). Apart from the breakage dissipation due to grain crushing, energy may be lost during shearing due to frictional dissipation. To account for this shear‐related dissipation at the macroscale, we can either adopt the generalized Hoek‐ Brown failure criterion (equation (9)) (without considering any cohesion) or the Mohr‐Coulomb failure criterion (equation (10)). qf ¼ M *pr pf =pr n

ð9Þ

qf ¼ Mpf

ð10Þ

where M and M* are friction‐related material constants; n is the Hoek‐Brown nonlinearity constant (0.5 to 1) and pr is reference pressure (1 KPa). Subscript f in equations (9) and (10) indicates stress state at failure. If n goes to 1, the cohesionless Hoek‐Brown failure criterion reduces to Mohr‐ Coulomb type failure criterion which was used by Einav [2007c]. [17] The combination of those failure criteria (equations (9) or (10)) with breakage yield criterion (equation (8)) gives yield conditions (equations (11) or (12)) in mixed stress‐ energy space (for models based on Hoek‐Brown and Mohr‐ Coulomb criteria, respectively): y¼

For simplicity, we use linear elasticity for the present breakage models. However, the breakage formulation is sufficiently general to accommodate any forms of elasticity, e.g., pressure‐dependent elasticity as illustrated by Einav [2007b] and Nguyen and Einav [2009]. From equation (3), the following stress‐strain‐breakage relationship and breakage energy EB are obtained:

ð7Þ

2 ð1 BÞ2 EB q þ 1 0 n Ec M *p1n r p

ð11Þ

2 ð1 BÞ2 EB q þ 1 0 Ec Mp

ð12Þ

y¼

[18] The breakage model based on Hoek‐Brown failure criterion is used for comparison purpose in some situations (wherever indicated), while most of the analysis in this study is based on its Mohr‐Coulomb type counterpart. Figure 1 depicts the breakage yield surfaces in p‐q‐B space with account given to equation (6). [19] We note that besides the energy loss due to frictional shear, the dissipation from grain rearrangement/friction, following a crushing event, can occur even in isotropic loading conditions due to pore collapse [Einav, 2007b; Nguyen and Einav, 2009]. This kind of coupling between breakage and volumetric plastic strain is accounted for through the introduction of the coupling angle w [Einav, 2007b] that governs

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Figure 1. Typical yield surface in q‐p‐B space for different failure criteria, (a) Mohr‐Coulomb; (b) Hoek‐Brown.

the effects of pore collapse on the material behavior. Further details on w can be seen in section 3.2. The evolution laws for breakage and inelastic strains (for Mohr‐Coulomb and Hoek‐Brown type shear failures, respectively) can be written as (dl is the common nonnegative multiplier; d ij is the Kronecker delta): ð1 BÞ2 cos2 ! ; Ec

ð13Þ

ð1 BÞ2 EB sin2 ! and pEc

ð14Þ

B ¼ 2

"pv ¼ 2

"ps ¼

8 q > > < 2 M 2 p2 > > : 2

[21] Using equations (7) and (17), the stress increment can be expressed: ij ¼ ð1 #BÞDijkl "kl ð1 #BÞDijkl Qkl þ

#ij B ð18Þ ð1 #BÞ

From the consistency condition of the yield function (equations (11) or (12)) we have: y ¼

@y @y @p @y @q B þ ij þ ij ¼ 0 @B @p @ij @q @ij

ð19Þ

Substitution of dsij from equation (18) in (19) results in:

Mohr-Coulomb failure criterion q

2 M *2 ðpr ð p=pr Þn Þ

y ¼ ð1 #BÞYij Dijkl "kl þ

Hoek-Brown failure criterion

@y Yij Xij B ¼ 0 @B

ð15Þ where Xij ¼ ð1 #BÞDijkl Qkl þ

The flow rules above can be expressed in tensorial forms, which will later be needed for the derivation of the tangent stiffness tensor:

"pij ¼

and Yij ¼

! 8 > 3sij ð1 BÞ2 EB sin2 ! ij > > þ > < 2 pEc 3 M 2 p2

#ij ð1 #BÞ

@y @p @y @q þ : @p @ij @q @ij

Mohr-Coulomb failure criterion

> 3sij ð1 BÞ2 EB sin2 ! ij > > > þ : 2 3 M *2 ðpr ð p=pr Þn Þ2 pEc

2.3. Formulation of Tangent Stiffness [20] The formulation of the tangent stiffness tensor Lijkl, which will be used for the localization analysis, is briefly presented in this section. Combining equation (13) and (16), we define:

ð20Þ

!

ð16Þ Hoek-Brown failure criterion:

The breakage increment is obtained from equation (20) as, B ¼

ð1 #BÞYij Dijkl "kl : Yij Xij @y=@B

ð21Þ

The stress increment can then be rewritten as "pij Qij ¼ B

ð17Þ

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ij ¼ Lijkl "kl ;

ð22Þ

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Table 1. Constitutive Parameters of Bentheim Sandstone Parameters

a

G K M or M* Ec n w

Mohr‐Coulomb

Hoek‐Brown Type

7588 MPa 13833 MPa 1.7 4.67 MPa ‐ 70°

7588 MPa 13833 MPa 210 5.41 MPa 0.6 70°

a Furthermore, # = 0.85. Distinguished from the above parameters, the # is a grading index, disconnected from stress‐strain data.

where the elastic‐plastic‐breakage tangent stiffness tensor Lijkl is, Lijkl ¼ ð1 #BÞDijkl

ð1 #BÞXij Ymn Dmnkl : Yij Xij @y=@B

ð23Þ

The modulus tensor (equation (23)) can be described in terms of material shear and bulk modulus (G and K, respectively), slope of yield locus (m), dilatancy factor (b p) and stress state (p, q) at the crushing inception (B = 0) as,

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sandstone, which exhibits significant compactive behavior and distinct compaction bands in experiments. Moreover, this rock is nearly monomineralic (95% quartz, 2% orthoclase and 3% kaolinite). The model capability in capturing the behavior of other sandstones and the evolving gsd due to grain crushing has been demonstrated by Nguyen and Einav [2009] and Nguyen et al. [2008]. The application of the current analysis to these sandstones is therefore straightforward, including the model assessment and localization analysis. Furthermore, the connections between the internal variables and their underlying micromechanisms will be explored in a parametric study. 3.1. Model Parameters and Response [24] All parameters for the above breakage model were obtained from the experimentally observed stress strain response and grain size distribution of Bentheim sandstone [Wong et al., 2001; Baud et al., 2004]. The stress‐strain response in elastic regime was used to calibrate the elastic moduli (K and G), while the critical breakage energy Ec was determined from the following formulation EC = P2cr#/2K [Einav, 2007c], in which the (isotropic) crushing pressure

Lijkl ¼ Dijkl

1 ¼

3GNij þ p Kij þ #ij 1 cos2 ! ð3GNkl þ Kkl Þ #ðq pÞ1 cos2 ! þ 6EB G21 cos2 ! =ð#q1 þ 6GÞ þ 3G þ p K

8 2 2 < M p =qEc :

for Mohr-Coulomb failure criterion

2n M *2 p2r pf =pr =qEc

for Hoek-Brown failure criterion

and (see also the flow rules in section 2.2) p ¼ "pv ="ps

ð25Þ

[22] In the above expressions, the definitions of the yield parameter m and dilatancy factor b p are identical to those used in previous works [e.g., Rudnicki and Rice, 1975; Wong et al., 1997; Issen and Rudnicki, 2000]. In the current breakage model these parameters also have links with other physical quantities governing the response of the model. In particular, m is governed by the shape of the yield surface, which through the breakage mechanics theory is a direct consequence of the assumption that the grain crushing dissipation is equal to the loss in the residual breakage energy; bp depends on the relative amount of dissipation due to grain crushing and pore collapse, through the coupling parameter w (see evolution rules for breakage and plastic strains in section 2.2).

3. Model Assessment [23] In this section the model is assessed against available experimental data. This is essential to obtain meaningful model parameters related to real material behavior, before examining the localization features of the model. Because we focus on the formation of compaction bands in porous rocks, we selected the highly porous (23%) Bentheim

where Nij ¼ sij =q;

ð24Þ

(Pcr) was obtained from the isotropic compression. According to experimental findings [Wong et al., 2001; Baud et al., 2006] for Bentheim sandstone this crushing pressure Pcr varies from 390 MPa to 420 MPa. It is important to note that the above range of Pcr were experimentally obtained from both dry [Wong et al., 2001; Baud et al., 2004] and wet [Baud et al., 2006; Tembe et al., 2008] Bentheim sandstone samples. The parameter M, governing the frictional shear, was determined from the ratio of ultimate shear stress (qf) to mean stress ( pf) from any conventional triaxial tests. For the grading index #, equation (2) was used which requires information on the initial and ultimate gsd’s. From experiments [Schutjens et al., 1995] we found that the grain size distribution is initially very narrow, having maximum grain diameter of 550 mm and minimum grain diameter of 50 mm. We assume that both the initial (g0(d)) and the ultimate (gu(d)) gsds are of power laws: g 0 ðd Þ ¼

ð3 0 Þd 20 ð3 u Þd 2u ; gu ðd Þ ¼ 3u 30 30 dM dm dM dm3u

ð26Þ

where dm is the minimum grain size, dM is the maximum grain size and a0 and au are the power law coefficients. We use a0 = −4 to reflect a narrow initial gsd; this is in agreement with the observation by Schutjens et al. [1995]. For the ultimate fractal gsd the fractal dimension is taken as au = 2.7. This fractal dimension for the ultimate gsd is well inside the proposed range of 2.5–2.8 [Sammis et al., 1986]. The remaining parameter, the coupling angle w, was determined from the slope of inelastic stress‐strain response. All the required parameters obtained for the breakage constitutive model in this study are listed in Table 1.

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Figure 2. Numerical (using the model described in section 2) and experimental [Baud et al., 2004] comparisons of stress‐strain responses of Bentheim sandstone under drained triaxial loading at different confining pressures; (a) mean stress versus volumetric strain; (b) differential stress against axial strain. [25] The model responses are plotted in Figure 2. Good agreement between numerical predictions and experimental observations can be seen, thus confirming the above parameter calibration. We can see that the breakage model based on linear elasticity works well in this case for Bentheim sandstone, although in general pressure‐dependent elasticity may be required [Nguyen and Einav, 2009]. [26] The use of a limited number of physically identifiable and determinable parameters and the agreement between experimental observation and numerical prediction highlight the importance of first principles in developing constitutive models. We note that the model was designed to tackle high‐pressure behavior due to grain crushing effects. The incorporation of cementation effects and the modeling of cement failure to deal with low‐pressure behavior within the framework of breakage mechanics would require resorting

to the micromechanics of bonded grains, rather than adding any fitting parameters. This enhancement of the theory is being carried out, with promising preliminary results. A successful example of a further development of breakage mechanics theory to model grain crushing in unsaturated granular materials, based solely on first principles at the grain scales, has been demonstrated in a recent work by Buscarnera and Einav [2011]. 3.2. Effect of gsd and Pore Collapse on Material Constitutive Behavior [27] In breakage mechanics theory, the grading index # (equation (2)) depends on the initial and ultimate gsd’s of the granular materials. The ultimate gsd can be conveniently assumed to be fractal capped by an ultimate minimum particle size and maximum particle size, and a reasonable

Figure 3. Stress‐strain responses of Bentheim sandstone; (a) varying # with constant w; (b) varying w with constant #. 6 of 14

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Figure 4. Schematic representation of the orientation of a localization band within a cylindrical sample. fractal dimension [Einav, 2007a]. It represents a fixed point that attracts the evolution of current gsd [Ben‐Nun et al., 2010]. On the other hand, the initial gsd depends on the initial microstructure of the material, which is essentially controlled by long‐term processes of healing and granulation [e.g., Marone et al., 1995; Renard et al., 2000]. If the initial gsd is close to the ultimate gsd (i.e., smaller # for well sorted gsd) higher external energy is required to crush the material compared to those having higher # [Einav, 2007b]. This shows the important role of the grading index # in governing the crushing potential of the granular material. Figure 3a illustrates the effect of # on the volumetric stress‐ strain behavior of a typical sandstone, while keeping the other model parameters fixed. It can be seen that the inelastic response is becoming softer with increasing #, consistent with its notional role for crushing potential. [28] The effect of pore collapse arising passively from fragment rearrangement, is presented in Figure 3b, simply by varying the parameter w. We note that this parameter only implicitly characterizes the role of the initial porosity and intergranular contact friction in governing the material response: higher porosity (of both the grains and their representative volume element) and lower grain contact friction would result in higher stress drop following a crushing event. As an example of this effect let us consider the case of two materials with identical gsd, and hence same #. If, however, one material contains microporous grains but not the other, it is straightforward to demonstrate that the materials with microporous grains must suffer more porosity reduction than the other one. The volumetric plastic strain is the internal variable representing these effects in the current breakage model. An explicit link between this plastic strain, its associated coupling parameter w, and the actual porosity reduction is not provided in this study. However, the reader can refer to a more sophisticated model in which the material porosity acts as an internal variable of the continuum model [Rubin and Einav, 2011]. According to equations (13) and (16), decreasing w signifies the increase in breakage dissi-

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pation, which indicates the dominance of dissipation due to grain crushing over the grain rearrangement and intergranular contact friction [Einav, 2007b]. On the other hand the volumetric plastic dissipation dominates the failure process if the coupling angle w is higher. The maximum value of w is 90° which implies the limit of zero breakage increment (equation (13)) and total inelastic dissipation that is purely plastic. Therefore the macroscopic material response is elastic perfectly plastic, as shown in Figure 3b. Any intermediate value of w between 0°–90° denotes a combined elastic breakage plastic response. [29] The roles of the material initial microstructures in governing the response of the material model can be clearly seen in Figure 3, with reference to the changes in their microstructural parameters # and w. This unique feature distinguishes the present breakage model from many existing plasticity‐based models in the literature. A hardening parameter, as usually required in these plasticity‐like models, is not necessarily derived here; this is because the hardening behavior of the model is directly governed by the grading index # and pore collapse parameter w, besides other elasticity properties.

4. Condition of Localization [30] The presence of local heterogeneity/weak zone at microscopic level produces stress concentration during the deformation process. Therefore the deformation behavior of granular rock material under some circumstances can bifurcate from homogeneous to inhomogeneous localization. In such a case, a localization band is observed at macroscopic level. We use the discontinuous bifurcation condition [Rudnicki and Rice, 1975] to detect the formation of compaction band. The classical discontinuous bifurcation condition is: _ ¼ 0: n Li Lo : e_ o þ ðAÞ m

ð27Þ

where n is the band orientation vector (see Figure 4); Li (equation (23)) is the tangent stiffness tensor inside the localization zone; Lo and e_ o are the tangent stiffness and strain rate tensor outside the localization zone, respectively; m is the orientation of relative velocity between opposite sides of the localization band; A = n · Li · n is the localization tensor, also termed the acoustic tensor. [31] Equation (27) is usually simplified considering identical tangent modulus tensors inside and outside the localization band [Rudnicki and Rice, 1975; Issen and Rudnicki, 2001; Chemenda, 2007]. Therefore the modified bifurcation condition is, n Li n ¼ jAj ¼ 0

ð28Þ

[32] This assumption, however, only holds either for incrementally linear models [Chambon et al., 2000] or if there are constraints over continuously evolving strain rates [Neilsen and Schreyer, 1993]. Due to the fact that the tangent stiffnesses of the material inside and outside the band are different in the case of our breakage model, the condition

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for discontinuous bifurcation is [Rudnicki and Rice, 1975; Chambon et al., 2000], n Li n ¼ jAj 0:

stress and strain tensors for 1‐D compression are presented in matrix form, 2

61

6 ij ¼ 6 6 0 4

ð29Þ

[33] The normal vector n of the localization band is illustrated in Figure 4. This normal vector can be expressed as follows, 8 9 8 9 < n1 = < cos cos = ni ¼ n2 ¼ sin : : ; : ; n3 cos sin

2

ð31Þ

5.1. The 1‐D Compression Test [35] We consider only pure compaction band (a = ±p/2) in this case. Therefore the normal vector (equation (31)) reduces to ð32Þ

Substituting equation (32) into (29) yields the bifurcation condition: L3333 0:

1

3

7 7 07 7; 5

ð34Þ

0 0 0

K þ 4G=3

3 7 7 7 7 7 5

ð35Þ

where n is the Poisson’s ratio and s (= s33) is the principal compressive stress at the onset of yielding. L3333 is determined from equation (24) as

ðK þ 4G=3Þðtan2 ! þ 2 2 Þ 2 4G 2 sec2 ! 3K tan2 ! þ ð1 2 Þ þ 2 1 tan2 ! þ 2 þ1 ð1 2 Þ 4G # 3K

[34] The breakage model described in section 2 and the localization condition (equation (29)) are used in a theoretical analysis of compaction band formation in porous rocks. Two different loading conditions are taken into account: 1‐D compression and triaxial drained loading. The formation of only pure compaction band is assumed in the 1‐D compression case to make it analytically tractable. Nevertheless this gives us an understanding of the micromechanics of compaction band formation in this simplest case. Similar assumption and analysis have also been used by Chemenda [2009]. On the other hand, drained triaxial loading condition is chosen because it is the most widely used loading path for material characterization. Due to complicacy of the model we perform the entire analysis numerically for this case. The results of localization analysis using the above mentioned loading conditions along with discussions are presented in sections 5.1 and 5.2.

0 1 gT :

0

L3333 ¼ ð K þ 4G=3Þ

5. Theoretical Analysis of Compaction Band Formation

ni ¼ f 0

0 0

6 6 6 "ij ¼ 6 0 0 6 4 0 0

ð30Þ

For axisymmetric cases, equation (30) can be simplified to have the following form: 8 9 8 9 < n1 = < 0 = ni ¼ n2 ¼ sin : : ; : ; n3 cos

0

0

ð33Þ

The condition of compaction localization is determined at the onset of yielding/crushing (B = 0). The material is loaded in 1‐D compression from zero stress state. The corresponding

ð36Þ where 2 ¼ 4G2 =K 2 M 2 :

From equation (33) and (36) we obtain the critical grading index #cr for the onset of pure compaction: "

#1 2 tan2 ! 4G 22 sec2 ! 3K #cr ¼ 2 2 þ 1þ 1þ : 1 2 3K 4G ð1 2 Þ2 ð37Þ

[36] The variation in the critical grading index (#cr) is plotted in Figure 5 against the Poisson’s ratio, for different values of M and w. It shows that the critical grading index (#cr) required for the compaction localization decreases with increasing pore collapse parameter (w), and vice versa. In other words, according to the current formulation the compaction band formation arises from the combined effects of grain crushing and pore collapse. Any reduction in one contributor’s effects indicates a different initial microstructure, and hence requires the compensations from the other to maintain the same probability for compaction band formation. Our analysis shows that crushable granulated rocks having larger #, i.e., those that initially consist of uniformly graded particles, should be more susceptible to compaction localization. From the micromechanical point of view, the potential for particle crushing in such a case is smaller than those having higher # [Einav, 2007a, 2007b]. This results in a stiffer, and less compacting, macroscopic response (Figure 3a). Therefore in such a case, the effects of pore collapse on the macroscopic behavior must be significant enough to maintain the same level of compacting response. Higher values of w are therefore required.

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Figure 5. Variation of #cr with Poisson’s ratio n, for various w and M in 1‐D compression. The line #cr = 0.9 is added as an example to highlight a typical uniformly graded with narrow gsd. Any point below this line would mark pure compaction for that material. 5.2. Conventional Drained Triaxial Test [37] The localization analysis at the onset of inelastic deformation was also carried out under drained loading using the discontinuous bifurcation condition (equation (29)). During loading the principal stresses maintain the following

condition: s3 > s1 = s2. Figure 6 represents the initial yield envelopes obtained from our theoretical models (for different failure criteria) and experiments [Baud et al., 2004]. We determine the set of stress state during the initial yielding for which compaction bands most likely occur. The

Figure 6. Initial yield envelopes and predicted stress states at the formation of compaction localization for Bentheim sandstone. (a) Mohr‐Coulomb; (b) Hoek‐Brown. 9 of 14

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Figure 7. Prediction of the normality condition in Bentheim sandstone. thick black line over the initial yield surface in Figure 6 denotes the numerically predicted zone where either shear enhanced or pure compaction bands may develop. The open circular symbols in brittle‐ductile transitional regime denote the experimentally observed stresses at the onset of shear enhanced compaction bands whereas the triangular solid symbols are the peak stresses during brittle failure [Baud et al., 2004]. In very high pressure regime no localization failure is observed. This may be manifested due to homogeneous deformation or diffuse compaction band formation over the entire sample under high confining stress. Similar damage distribution in delocalized manner in high‐pressure regime was also experimentally observed by Wong et al. [2001]. Furthermore, in our numerical test, shearing the material model beyond the initial yield surface and at relatively high pressure (e.g., up to 350 MPa), will eventually lead to compaction localization. This pressure is, however, smaller than the experimental maximum pressure (450 MPa) at which compaction localization can be observed [Baud et al., 2004]. As also numerically experienced, the closer to the isotropic compression line the loading path is, the easier the deformation would evolve into cataclastic flow without any compaction localization. [38] One can also note in Figure 6 that the breakage model with Hoek‐Brown type frictional criterion better fits the entire experimental yield surface than the one based on Mohr‐ Coulomb type failure criterion. This is because the Mohr‐ Coulomb critical state line (equation (10)) does not allow the breakage yield surface to match the experimental data at low confining pressure. We may need a more sophisticated model which gives the flexibility in critical state response, as suggested by Nguyen and Einav [2009]. The Hoek‐ Brown type dissipation mechanism could be a better alternative in this context due to the pressure‐dependent nature of the critical state line. [39] The experimental and theoretical inelastic compaction factors DFp/D"pa (≈ D"pv /D"pa, according to Wong et al. [1997]) are compared in Figure 7 for Bentheim sandstone. Baud et al. [2006] also made a comparative study between

the experimental and theoretical inelastic compaction factors using DiMaggio and Sandler cap model [DiMaggio and Sandler, 1971] and associative flow rules. However, this model predicts inelastic compaction factor higher than the observed experimental data. We show here that the experimental inelastic volumetric strains are in good agreement with the predictions by the breakage constitutive model. [40] Figure 8 highlights the contours of the determinant of the acoustic tensor (or the strain localization tensor) against the band orientation angle (a) and the slope of yield surface (m) for Bentheim sandstone using the breakage model associated with Mohr‐Coulomb failure criterion. The color bar in this plot indicates the values of the determinant of the acoustic tensor. In Figure 8 the innermost zone denotes negative determinant of the acoustic tensor and it is bounded by ∣Aij∣ = 0. According to the discontinuous bifurcation criterion (equation (29)) compaction localization is possible only within this zone. We already presented this similar zone in p‐q space through the thick black line in Figure 6. It is clear from Figure 8 that for any stress state, in terms of m, we get a set of orientation angles which are physically equally favorable in forming compaction or shear enhanced compaction band. In practice, localization band having orientation angle 0° ≤ a ≤ 10° can be classified as pure compaction band whereas those with 10° ≤ a ≤ 45° are treated as shear enhanced compaction bands [Baud et al., 2004]. Our analysis predicts that the band orientation falls within the range of 0° to 40° which is quite close to the experimental observations [Baud et al., 2004]. [41] The orientation angles corresponding to different stress states at which compaction localization occur are plotted in Figure 9, showing different modes of localization. The location and the size of the localization zone depicted on the initial yield surface are completely dependent on the stress state, grain size distribution and porosity of the respective material. Shear bands are observed at the transition of brittle and ductile failure. On the other hand at the right boundary of the localization zone, pure compaction bands are observed due to ductile nature of porous rocks under shearing at high

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Figure 8. Contours of the determinant of the acoustic tensor for Bentheim sandstone at crushing inception (B = 0). confining pressure [Klein et al., 2001]. Beyond this zone no localization is observed, due to homogeneous deformation under high confining pressure [Wong et al., 2001]. We note that our model aims at tackling high‐pressure behavior with grain crushing in porous rocks. Ongoing efforts are being carried out to equip the model with new capabilities to capture the material behavior at low confining pressure, where the formation of extension/dilation bands [Bésuelle, 2001] is expected. [42] We compare the predicted values of m and b p at the onset of localization against the experimental results in Figure 10. Increase in the absolute value of m and bp indicates transition from compaction to shear band. Our

Figure 9. Stress states and corresponding localization band orientations over initial yield surface (Bentheim sandstone).

predictions fairly agree with experimental observation. It is clear (Figure 10) that the predicted range of m is almost identical to its experimental counterpart. According to the experimental observation by Wong et al. [2001], m should be greater than −1.5 for compaction localization which is well predicted by our breakage model. The experimentally observed bp is higher than our prediction as shown in Figure 10. However, Figure 10 shows that enhancement to the model behavior (e.g., using Hoek‐Brown criterion) leads

Figure 10. Possible zone of compaction localization in m − bp space for Bentheim sandstone. (The values of m and b p calculated by some researchers [Issenpand Rudnicki, 2000; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Wong et al., 2001] werepinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p − ffi( = 1=2sij sij ) space. Due to the use of p–q (q = 3=2sij sij ) stress space in this study, appropriate adjustments were made for the presentation in p–q space.).

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Figure 11. Variation in zone of possible compaction localization for different # and w for Bentheim sandstone. to a better model prediction. Therefore improvements in the model’s capabilities may be a hint for better predictions that can help explain the current mismatch. This is the topic of our ongoing study. 5.3. Parametric Study [43] Given the links between the model parameters (# and w) and the initial microstructures (gsd and porosity) of the material (section 3.2), a parametric study is carried out to explore the effects of initial microstructures on the formation of compaction localization. Figure 11 shows the effects of initial gsd (through #) and pore collapse (through w) on the formation of compaction localization. As can be seen, the size of compaction localization zone strongly depends on the initial gsd and pore collapse effects. It is quite clear that the possible range of stress state (in terms of m) to trigger compaction localization gets narrower as w becomes smaller. We know that this parameter w reflects the effects of pore collapse on the material responses. In all cases, the dilatancy factor (bp) decreases (in absolute value) with decrease in w for any m, which indicates a reduction in inelastic volume change due lower pore collapse effects. Taking a close look at Figures 11a–11d, one can see the change in favorable stress state for compaction band formation due to the variation in #. The range of m − b p associated with localization

zone reduces with the decrease in # and vice versa. Increase in # indicates that the initial gsd is poorly graded and far away from ultimate gsd and therefore directly increases the chance of compaction localization. These numerical predictions confirm the analytical results in 1‐D compression case. [44] It can be seen in Figure 11 that the compaction localization regime extends to higher pressures (indicated by smaller values of m) when the grain structure is poorly sorted (high #) and/or the effects of pore collapse are high (high w). Material with well‐sorted grain structure and low porosity are therefore less likely to form compaction bands. The theoretical extreme cases totally precluding compaction localization are #→0 (initial and ultimate gsd’s coincide), and w→0 (very low internal porosity within the grains). The condition for no compaction localization, however, depends on the type of loading, and only in specific cases can an analytical expression be derived (see section 5.1). We can see from the trend in Figure 5 that for certain values w < 45° (low pore collapse effects) compaction localization can never be observed regardless of what the initial gsd is. [45] The formation of compaction localization, as shown in this paper, depends not only on the initial grain size distribution, but also on other features of the microstructure (e.g., initial porosity). From the experimental point of view,

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another factor contributing to the formation of compaction localization could be the stress distribution in rock samples, which is not always perfectly uniform, but depends on both the material microstructure and the specimen geometry. It is therefore possible that compaction localization can develop in relatively homogeneous samples, as numerically observed by Katsman et al. [2005] and Wang et al. [2008], and experimentally observed by Louis et al. [2007].

6. Concluding Remarks [46] We have presented a novel continuum modeling approach to capturing the micromechanics of compaction band formation in porous rocks. This model possesses explicit links between the evolution of the gsd and the macromechanical behavior of porous rocks. Our predictions are in agreement with experimental observations in terms of stress strain behavior of porous rocks, yield locus, favorable stress states for compaction localization and compaction band orientations. It also explains the micromechanics behind the formation of compaction bands. The role of the initial gsd, via the grading index #, is found to be a key factor for localization deformation. Any change to this initial grading would affect the range of favorable stress state for compaction band formation. On the other hand, the parameter w, as an indicator of pore collapse effects on the material response, controls both the favorable stress states for compaction localization and the inelastic volume change during localization deformation. [47] At this point we would like to emphasize the important feature of the bifurcation analysis thus presented. We have employed the general bifurcation condition by Rudnicki and Rice [1975] (R&R), applicable to any constitutive models. Our results have been presented using the usual yield parameter m and dilatancy factor bp as suggested in R&R. However, in the proposed breakage model these parameters are strongly linked with the actual driving physical mechanisms. In particular, the shape of the yield surface represented by m is here a direct consequence of the assumption that the grain crushing dissipation is equal to the loss in the residual breakage energy; b is linked with the competition between the dissipations due to grain crushing and pore collapse. These two connections underpin equations (23)–(24) and (36)–(37). These features are usually missing in plasticity‐ based models for geomaterials. [48] This simple approach provides a good understanding of the underlying mechanisms governing compaction localization. It still requires further improvements and investigations, which are an ongoing research in our group toward a better understanding of the complex intrinsic micromechanics behind compaction localization in porous rocks. [49] Acknowledgments. Arghya Das wishes to thank the University of Sydney International Scholarship scheme. Giang Nguyen and Itai Einav would like to acknowledge the Australian Research Council for the Discovery Projects funding scheme (projects DP110102645 and DP0986876). Support to Giang Nguyen through the University of Sydney Postdoctoral Fellowship scheme is also gratefully acknowledged. Finally, the authors would like to thank the two reviewers, Teng‐fong Wong and Patrick Baud, for their constructive comments that helped us improve the clarity and consistency of the paper.

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