dilatant hardening of fluid-saturated sandstone...dilatant hardening of ﬂuid-saturated sandstone...

Dilatant hardening of fluid-saturated sandstoneRoman Y. Makhnenko1,2 and Joseph F. Labuz1

1Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, Minneapolis, Minnesota, USA, 2Now atSoil Mechanics Laboratory - Chair “Gaz Naturel” Petrosvibri, School of Architecture, Civil and Environmental Engineering,Swiss Federal Institute of Technology, Lausanne, Switzerland

Abstract The presence of pore fluid in rock affects both the elastic and inelastic deformation processes,yet laboratory testing is typically performed on dry material even though in situ the rock is often saturated.Techniques were developed for testing fluid-saturated porous rock under the limiting conditions of drained,undrained, and unjacketed response. Confined compression experiments, both conventional triaxial and planestrain, were performed on water-saturated Berea sandstone to investigate poroelastic and inelastic behavior.Measured drained response was used to calibrate an elasto-plastic constitutive model that predicts undrainedinelastic deformation. The experimental data show good agreement with the model: dilatant hardening inundrained triaxial and plane strain compression tests under constant mean stress was predicted and observed.

1. Introduction

The presence of pore fluids in rock can affect both the elastic and inelastic response [Rice, 1975; Rice and Cleary,1976; Detournay and Cheng, 1993]. However, laboratory testing and material characterization, specifically thoseresponsible for inelastic behavior, are typically performed on dry rock. The inelastic deformation of brittlegeologic materials is often inhibited by an increase of mean stress (compression positive). Thus, an increase inpore fluid pressure decreases the effective mean stress and promotes inelastic deformation; conversely, adecrease in pore fluid pressure tends to inhibit inelastic deformation. Coupling of deformation with porefluid diffusion also introduces time dependence into the response of an otherwise rate-independent solid.

It is convenient to consider two limiting timescales of fluid-saturated solid response. If the timescale ofdeformation is rapid in comparison to that for diffusion so that the fluid mass in a material element remainsconstant, then the response is termed “undrained.” The long-time or “drained” response is the one for which thelocal pore fluid pressure is constant. Both elastic and inelastic material parameters differ under drained andundrained conditions.

Depending on the stress state, the constitutive response of brittle rock can involve dilation (incremental volumeincrease) as deformation becomes inelastic. When the rock is fluid-saturated and the timescale does not allowdrainage, suction is induced in the pore fluid, and by the effective stress principle, the rock is dilatantly hardenedover the resistance that it would exhibit to a corresponding increment of drained deformation [Rice, 1975].

The characterization of inelastically deforming rock under well-controlled drainage conditions is required inorder to understand the coupling between rock deformation and pore fluid diffusion just before failure[Nur, 1972; Rice and Cleary, 1976; Rudnicki, 1985] and to characterizemechanical behavior of partially molten rock[Renner et al., 2000]. The problem of stress concentration near a passing rupture front along a fault in afluid-saturated rock is effectively undrained on a short timescale; if the material is plastically dilating, theundrained deformation is found to be more resistant to shear localization than predicted by neglect of porepressure changes [Viesca et al., 2008]. These considerations are particularly important when analyzing thestability of long-term underground storage for the disposal of nuclear waste or carbon dioxide. For example,when the injection rates were increased at a CO2 sequestration site, a significant increase in microcrackingactivity was observed [Oye et al., 2013], possibly due to the effective mean stress being reduced by increasedpore pressure so the response of the host rock was inelastic and undrained. Knowledge of dilatant behavior isalso important in oil production, where the permeability of a reservoir can change due to inelastic deformation[Yale et al., 2010].

Drained and undrained behavior of sand has been investigated by a number of researchers [Han andVardoulakis, 1991; Mokni and Desrues, 1998] and the dilatant behavior of water-saturated dense granular

MAKHNENKO AND LABUZ ©2015. American Geophysical Union. All Rights Reserved. 909

PUBLICATIONSJournal of Geophysical Research: Solid Earth

RESEARCH ARTICLE10.1002/2014JB011287

Key Points:• Fluid-saturated sandstone was testedunder drained and undrainedcompression

• Material deformation was characterizedby an elasto-plastic constitutive model

• Model calibrated with measureddrained response predictedundrained behavior

Correspondence to:R. Y. Makhnenko,[email protected]

Citation:Makhnenko, R. Y., and J. F. Labuz (2015),Dilatant hardening of fluid-saturatedsandstone, J. Geophys. Res. Solid Earth,120, 909–922, doi:10.1002/2014JB011287.

Received 14 MAY 2014Accepted 13 JAN 2015Accepted article online 17 JAN 2015Published online 26 FEB 2015

http://publications.agu.org/journals/http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)2169-9356http://dx.doi.org/10.1002/2014JB011287http://dx.doi.org/10.1002/2014JB011287

materials has been characterized. The limitations in these experiments are that water is usually assumed tobe incompressible and the undrained response is reported to be isochoric, which is not the case for rock,where bulk modulus of thematerial is comparable, if not larger, than that of the pore fluid. At the near-surfacerange of mean stresses, rock-like materials exhibit volume strain less than 1% prior to failure in both drainedand undrained compression tests [Makhnenko et al., 2012] and accurate deformation measurements arerequired for the proper assessment of the material response.

Only a few experiments document dilatant hardening behavior of rock. Brace and Martin [1968] investigateddrained response of saturated sandstone and granite under conventional triaxial compression. In experimentsat loading rates greater than a critical value (>10�4 s�1 for Pottsville sandstone and >10�7 s�1 for Westerlygranite), the rock was as much as 50% stronger than at constant pore pressure. It was explained by the fact thatpore pressure did not have enough time to equilibrate in the dilating rock specimens; hence, the effectiveconfinement increased, which augmented the peak shear stress. Bernabé and Brace [1990] reported onconventional triaxial compression tests on silicone-saturated Berea sandstone. For specimens tested at 10MPaeffective confinement, the peak stress was 25% larger when the strain rate was increased from 2×10�4 to2× 10�3 s�1, but the same increase in the strain rate at larger effective confinements (70 and 120MPa) in theductile regime caused a significant decrease (13–17%) in rock strength. Schmitt and Zoback [1992] performedrupture tests on internally pressurized, thin-walled hollow cylinders of Westerly granite. For the tests onsaturated specimens, where the inner cavity pressure was increased rapidly with respect to the diffusivity, thesubstantially (up to 50%) increased apparent tensile strengths and deformation moduli were measuredcompared tomore slowly pressurized tests. The authors believed that the diminished pore pressures provided apossible reason for the less compliant response in rapidly pressurized tests, but it was not measured directly.

Sulem and Ouffroukh [2006] suggested an elastoplastic constitutive model with stress-dependent elasticityand damage, which described the inelastic response of a fluid-saturated rock; the model was validatedthrough back analysis of drained and undrained tests on Fontainebleau sandstone. Lei et al. [2011] conductedconventional triaxial tests on Berea sandstone with the same initial confinement and pore pressure underdrained and undrained conditions and observed that the peak axial stress in the latter was 30% larger. Dudaand Renner [2013] performed conventional triaxial compression experiments on three fluid-saturatedsandstones. By changing the rate of loading from 10�7 to 10�3 s�1, approximate drained and undrainedconditions were achieved. For the tests conducted with the same initial conditions, the strengthening (morethan 50%) of an effectively undrained (fast loading) specimen was observed. However, the inelastic responsecould not be characterized properly because of limitations in strain and pore pressure measurements. Theinterpretation of the observed hardening under high loading rates was that the propagation and extensionof microcracks were associated with dry conditions. Pore pressure at the crack tips could not be maintainedand effective mean stress increased, thus delaying microcrack propagation and development of faulting,which is similar to the explanation given by Rudnicki and Chen [1988].

Generally, studies on dilatant hardening deal with applying different strain rates and drained boundaryconditions and observing hardening when deformation proceeded too fast to maintain effective drainage. Inthis kind of test, pore pressure does not have enough time to equilibrate and such conditions could be calledquasi-undrained, because mass of the fluid in the specimen is not preserved constant, but the rate of itschange is slower than the rate of loading. However, this creates a variation of pore pressure throughout thespecimens, making it difficult to evaluate effective stress and parameters that depend on it. Thus, theconstitutive response of the rock under different drainage conditions could not be characterized.

This paper deals with drained and undrained testing of a porous rock, Berea sandstone, under constant andrelatively slow (~10�6 s�1) strain rates, which allow pore pressure equilibration in the specimens. The limitingconditions are achieved by maintaining constant pore pressure in the drained experiment and no-flowboundary conditions for the undrained one. The University of Minnesota plane strain apparatus was modifiedfor testing of water-saturated specimens under both plane strain and axisymmetric compression configurations.The device allows development of three different principal stresses and measurement of principal strains, aswell as the ability to apply and control pore pressure within the rock and monitor acoustic emission activity.The emphasis is on accurate measurements of poroelastic, friction, and dilatancy characteristics of the materialfrom the experimental data. The dilatant hardening response is presented then in terms of the parametersformulated by Rice [1975] and the constitutive model predictions are compared with the test results.

Journal of Geophysical Research: Solid Earth 10.1002/2014JB011287


2. Constitutive Model

The constitutive model chosen for describing the inelastic response of fluid-infiltrated rock was formulatedby Rice [1975] for the plane strain deformation state of confined shearing. It illustrates the hardening effect ofdilatancy-induced pore pressure reduction. The model was reformulated by Rudnicki [1985] in terms ofporoelastic parameters for rock subjected to pore pressure p, shear stress τ, and mean (hydrostatic) stressP= (σ1 + σ2 + σ3)/3 and is reviewed here.

For stress increments dτ, d(P� p) that involve continued inelastic deformation, the increments of shear straindγ and volume strain dε (compression positive) are

dγ ¼ dτ=Gþ dγp (1)dε ¼ dP=K þ dεp (2)

where dγp and dεp are the increments of inelastic shear and volume strain, respectively, G is the shearmodulus, and K is the bulk modulus. An increase in mean stress inhibits inelastic deformation in brittle rock.Thus, the inelastic increment of shear strain takes the form

dγp ¼ dτ � μdPð Þ=H (3)where H is an inelastic hardening modulus (shown in Figure 1a for dP=0) and μ is a friction parameter, whichcan be measured as the local slope of the yield surface separating elastic and inelastic states in the plane ofτ and P� p (Figure 1b).For the case when there is no change in pore pressure during deformation (dp= 0), the so-called drainedresponse, and mean stress is constant (dP=0), dτ can be written as

dτ ¼ H1þ H=Gdγ (4)

The inelastic volume strain arises from processes that accompany inelastic shear and its increment is given byβ, the dilatancy factor (β< 0 for compaction):

dεp ¼ �β�dγp (5)

The sign convention here is compression positive, meaning that when the specimen dilates, its incrementalplastic volume strain is negative.

To include the effects of pore fluid pressure, the (total) mean stress is replaced by the effective mean stress.For elastic deformation, the effective mean stress is written in Biot’s [1941] form: P� αp, where α is the Biotcoefficient. For inelastic deformation arising from opening and sliding of microcracks with small contactareas, Rice [1977] has shown that the appropriate form of the effective mean stress is P′= P� p. Thisdeduction is consistent with experimental observations on inelastic properties of brittle rock [Brace andMartin, 1968]. Thus, the increments of shear and volume strain are

dγ ¼ dτ=Gþ dτ � μ dP � dpð Þ½ �=H (6)

dε ¼ dP � αdpð Þ=K � β dτ � μ dP � dpð Þ½ �=H (7)

Figure 1. Constitutive response of saturated rock (modified after Rice [1975]). (a) Shear stress τ–shear strain γ behavior underdrained conditions, with mean stress P and pore pressure p being constant. H is an inelastic hardening modulus and G is theelastic shear modulus. (b) Yield surface in the plane of effective mean stress and shear stress; μ is a friction coefficient.



Fluid mass content per unit volume mf for elasticdeformation can be expressed as ρf · υ, where ρf isthe fluid density and υ is the apparent fluid volumefraction [Biot, 1973]. The expression obtained for anelastic increment of υ is [Rice and Cleary, 1976]

dυ ¼ υ� υo ¼ � α dP � dpð Þ=K � υdp=K 00s (8)

where υo is its reference value in the unstressedstate, i.e., υo =ϕ, the initial porosity of the rock ifthe material is fully saturated, and Ks00 is theunjacketed pore bulk modulus.

The inelastic increase in apparent fluid volumefraction dυp was shown to be the plastic dilatancy:dυp=�dεp [Rice, 1977; Viesca et al., 2008], and thetotal change in fluid volume fraction is the sum of itselastic and inelastic increments. Then, from theequation for the fluid bulk modulus Kf : dρf /ρf=dp/Kf,the increment of fluid mass content per unit volumedmf can be written as

dmf=ρf ¼ υdp 1=Kf � 1=K 00s� �� α dP � dpð Þ=K � dε p (9)

For undrained response, the change in fluid mass content per unit volume is 0. Setting dmf= 0 in (9) andconsidering a constant mean stress condition yields

dp ¼ �βKeffHþ μβKeff dτ (10)

where Keff can be written as

1=Keff ¼ υ 1=Kf � 1=K 00s� �þ α=K (11)

Therefore, for dilatant inelastic deformation (β> 0), the pore fluid pressure tends to decrease. When (10) issubstituted into (6) with dP= 0, the result is

dτ ¼ Hþ μβKeffð Þ1þ Hþ μβKeffð Þ=Gdγ (12)

Because the hardening modulus has been augmented by the term μβKeff comparing to the drained case(equation (4)), the response is stiffer and the rock is said to be “dilatantly hardened” (Figure 2). Parameters μ,β, Keff, and H are stress dependent, but their development with effective mean stress can be evaluated andused to determine undrained response.

3. Experimental Methods3.1. Plane Strain Compression

The University of Minnesota plane strain apparatus [Labuz et al., 1996] allows application of three differentprincipal stresses to a prismatic (100×87×44mm) rock specimen and accurate measurement of deformationin corresponding directions (Figure 3). Major principal stress σ1 is applied in the axial direction by a servohydraulicload frame; the minor principal stress (cell pressure) σ3 is applied in the lateral direction. Axial and lateraldisplacements are measured by four linear variable differential transformers (LVDTs).

A plane strain condition (ε2 ~ 0) is achieved by “wedging” the specimen in a thick-walled steel cylinder, whichprovides passive restraint. The intermediate principal stress cannot be controlled, but it can be calculated

Figure 2. Dilatant hardening due to undrained conditions inthe shear stress τ–shear strain γ diagram (modified after Rice[1975]). Undrained response under constant mean stress Pappears to be stiffer than its drained counterpart, because thehardening modulus H is augmented by the term dependenton friction coefficient μ, dilatancy factor β, and poroelasticmodulus Keff.



from the measured frame deformation andgeneralized Hooke’s law for an isotropic, elasticsolid [Makhnenko and Labuz, 2014]:

σ2 ¼ Eε2 þ ν σ1 þ σ3ð Þ (13)

Young’smodulus E and Poisson’s ratio ν aremeasuredin plane strain compression with constant σ3:

E ¼ Δσ1 Δε1 � 2Δε3Δε1 � Δε3ð Þ2

(14)

ν ¼ �Δε3Δε1 � Δε3 (15)

For plane deformation, it is convenient to introduce(i) average stress s, (ii) shear stress τ, (iii) volumestrain ε, and (iv) shear strain γ:

s ¼ σ1 þ σ3ð Þ=2τ ¼ σ1 � σ3ð Þ=2

(16)

ε ¼ ε1 þ ε3γ ¼ ε1 � ε3

(17)

Generalized Hooke’s law for plane strain becomes

Δγ ¼ 1GΔτ (18)

Δε ¼ 2 1þ νð Þ3K

Δs (19)

The apparatus allows the upstream pressure in the system to be controlled and the downstream pressureto be measured. Applying drained (Δp=0) and undrained boundary (Δmf= 0) conditions, the respectiveYoung’s moduli E and Eu and Poisson’s ratios ν and νu can be measured. Hence, drained K and undrainedKu bulk moduli can be calculated:

K ¼ E3 1� 2νð Þ (20)

Ku ¼ Eu3 1� 2νuð Þ (21)

Furthermore, because effective shear stress is equal to the total shear stress, the drained and undrained shearmoduli, G and Gu are equal:

G ¼ E2 1þ νð Þ ¼

ΔτΔγ

¼ Gu (22)

Skempton coefficient B is calculated from measurements of pore pressure increase Δp due to appliedincrement in mean stress ΔP, which in this case is induced by applying equal increments in major and minorprincipal stresses Δσ1 =Δσ3, and Δσ2 is calculated by taking Δε2≈ 0 in equation (13). After the contributionof the extra fluid volume in drainage lines VL over the total specimen volume V is considered, B can beobtained from [Bishop, 1976; R. Y. Makhnenko and J. F. Labuz, 2013]

B ¼ 12 1þνuð Þ Δσ1þΔσ3ð Þ

3Δp

h i� VLV KαKf

(23)

3.2. Conventional Triaxial Compression

Another simple geometry, which allows direct measurements of applied principal stresses and correspondingstrains, is a cylindrical specimen (height=104mm, diameter=51mm) tested in conventional triaxial compression.

Figure 3. Principal stresses and corresponding strains in theplane strain experiment. Major principal stress σ1 is appliedby a servohydraulic actuator, minor principal stress σ3 is cellpressure, and the intermediate principal stress σ2 is inducedby a stiff frame, which acts as a passive restraint and limits ε2to ~0.01 ε1. Prismatic (100 × 87 × 44mm) specimens areinstrumented with AE sensors; the upstream pressure iscontrolled and the downstream pressure is measured.



Here the major principal stress σ1 isapplied axially and intermediate andminimum principal stresses are equal toeach other σ2 = σ3 and applied by the cellpressure. Axial strain ε1= εa is measuredby strain gages and two LVDTs; tangentialand radial strains εθ and εr (equal foraxisymmetry) are measured by a pair oflateral strain gages: εθ = εr= ε2 = ε3. Apolyurethane membrane prevents thepenetration of the cell liquid (hydraulic oil)into the specimen and protects thestrain gages.

For this geometry, the following invariantscan be considered: mean stress P, deviatorstress q, volume strain εv, and shear strain εs:

P ¼ σ1 þ 2σ3ð Þ=3q ¼ σ1 � σ3ð Þεv ¼ ε1 þ 2ε3εs ¼ 2 ε1 � ε3ð Þ=3

(24)

Because application and measurement of the three principal stresses and corresponding strains arestraightforward in conventional triaxial compression, an unjacketed test can be conducted by applyingincrements of pore pressure equal to the increments in the mean stress: Δp=ΔP. Drained tests can beperformed by maintaining the pore pressure in the specimen constant; this condition is applied andpreserved by connecting the pore water lines to a microprocessor-based hydraulic pump that maintainswater pressure at a constant value, within a tolerance of 0.02MPa. By closing the inlet and drainagevalves and thus keeping the mass of the fluid in the specimen constant, undrained testing condition canbe achieved. The bulk moduli related to these three limiting states, i.e., unjacketed Ks′, drained K, andundrained Ku, can be measured by applying hydrostatic loading to a cylindrical specimen and recording thecorresponding change in volume strain. Measurements of Ks′ [R. M. Makhnenko and J. F. Labuz, 2013] andK then allow the calculation of Biot coefficient α:

α ¼ 1� K=K ′s (25)

Skempton coefficient B in triaxial compression is calculated frommeasurements of pore pressure increase Δpdue to application of hydrostatic mean stress ΔP=Δσ1 =Δσ3 and correcting for the extra fluid volume in thedrainage system (similar to equation (23)):

B ¼ 1Δσ1þ2Δσ3

3Δp

� �� VLV KαKf

(26)

Achieving full saturation is critical to guarantee that the rock was tested under drained or undrained conditions.The completeness of saturation, by which we mean here that all interconnected pores are filled with waterand no air is trapped there, is evaluated based on measurements of Skempton B coefficient [R. Y. Makhnenkoand J. F. Labuz, 2013]. It will be shown in section 4.2.3 that measurements of the introduced poroelasticparameters are sufficient for the calculation of effective bulk modulus Keff from equation (11), which plays a rolein characterizing the inelastic response of fluid-saturated rock.

3.3. Acoustic Emission

Failure of rock involves microcracking, which generates elastic waves known as acoustic emission (AE).For a wide variety of brittle rocks and over a wide range of confining pressures, dilatant deformation wasfound to be accompanied by small-scale fracturing [Scholz, 1968]. Thus, monitoring of AE activity in the rockcan be a tool for determining the onset of inelastic response [Riedel and Labuz, 2007]. Figure 4 represents

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Figure 4. Volume strain ε and total number of AE events versus shearstrain γ measured for the dry sandstone in plane strain compression.The AE rate significantly increased at the onset of inelastic response.



the results of a dry plane strain compressionexperiment on the sandstone in terms of volumestrain and recorded AE events. It can be seenthat the AE rate increases at the onset of inelasticdeformation, where the volume strain responsedeviates from linearity, and the AE rate isapproximately constant almost up to the peakload. The same technique was implementedfor fluid-saturated rock testing [Makhnenkoet al., 2012].

The high-speed data acquisition system(National Instrument model NI-5112 digitizer)of 8 bit analog to digital conversion resolutionhas four two-channel modular transientrecorders. The sampling rate was set at20MHz with a 0.2ms time window and 0.1mspretrigger. The AE system is triggered whenthe signal amplitude from a selected channelexceeded a preset threshold. The process iscontrolled by an in-house program writtenin LabView and data recording is almostcontinuous without interruption; because oftransducer resonance, a trigger hold of 10mswas used. This limited the AE rate to 100events/s, although the maximum observed ratewas 10 events/s.

The initial part of the material response inFigure 4 (and in similar figures) is related toseating effects. A small nonlinearity in machinestiffness contributes to the observed behavior[Makhnenko and Labuz, 2014].

4. Results and Discussion

Plane strain and conventional triaxial compression experiments were performed on water-saturated Bereasandstone specimens with uniaxial compressive strength UCS=42–44MPa, initial interconnected porosityϕ =0.23, and permeability k=40mD measured at 5MPa mean stress. This rock contains distinguishablebedding planes and exhibited slight (5%) anisotropy in compressional wave velocity. Tested specimens werecut from the same block and at the orientation such that themajor principal stresswas applied perpendicular tothe beds.

Axial strain at failure (peak shear stress) did not exceed 1.5% and the measurements of the poroelasticmaterial parameters [Makhnenko, 2013] allowed for the calculation of the critical strain rate ε̇cr [Renner et al.,2000]. Above this rate, effective internal drainage is lost and the material behaves as undrained; belowthis rate, drained (fluid-saturated) and dry rock exhibit the same strength characteristics [Brace andMartin, 1968]. For our experiments, the critical strain rate was approximately 5 × 10�1 s�1, which is thesame as the one predicted for water-saturated Berea sandstone by Renner et al. [2000] and at least 2 ordersof magnitude larger than the one observed by Bernabé and Brace [1990]. The axial strain rate in allexperiments was kept at a significantly smaller value of ~10�6 s�1 to promote equilibration of porepressure within the specimens.

4.1. Plane Strain Compression

After complete saturation of a sandstone specimen was achieved, plane strain compression test wasperformed. Axial load was applied under lateral displacement control and the cell pressure σ3 was preserved

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peak = 38.8 MPa

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Figure 5. Comparison of the drained (dark grey) and undrained(dark blue) plane strain compression experiments failed at differenteffective mean stresses: (a) stress paths on the shear stress–effectivemean stress diagram and (b) constitutive response on the shearstress–shear strain diagram. Effective mean stresses P′ at failurewere significantly different for the two tests.



constant. A series of drained and undrainedexperiments were conducted and theresults of two of them, one drained (darkgrey curve) and one undrained (dark bluecurve), are presented in Figure 5. Effectivemean stresses at failure, designated as P′,were significantly different for the tests(Figure 5b), although the initial effectivemean stress for each of them was similar:Po′=5.0–7.0MPa (Figure 5a). Stresses atfailure for drained and undrained specimenswere found to be in the good agreementwith the failure data from tests on dry Bereasandstone, so Terzaghi’s effective stressprinciple holds for this rock.

The drained test was performed withσ3 = 5.1MPa, initial pore pressurepo= 0.6MPa, and Po′=4.5MPa; theundrained test was conducted atσ3 = 10.0MPa, initial pore pressurepo=2.8MPa, and Po′=7.2MPa. It can beseen from Figure 5b that the undrainedresponse is “stronger” than its drainedcounterpart starting from a value ofγ=2.2 × 10�3. The effective mean stress atthe beginning of the undrained shearingwas slightly larger than that in the drainedtest. However, for τ =12MPa, drainedand undrained effective mean stressesbecome equal, and then further in loading,the undrained one becomes slightlysmaller than the corresponding P′ for thedrained test.

The increasing mean stress leads to the specimens’ compaction before they dilate, and their dilatancy isaccompanied by approximately constant AE rate in both drained (Figure 6a) and undrained (Figure 6b)experiments. It should be noted that undrained deformation of water-saturated soil is usually considered topreserve the volume of the specimen constant [Terzaghi, 1943], because the bulk modulus of a soil (order ofmegapascal) is much smaller than the bulk modulus of water (Kwater = 2.24 GPa), which means that watertakes most of the load. However, in the undrained testing of porous rock, its bulk modulus is on the order ofgigapascal and is comparable with the bulk modulus of pore fluid (Ku~ Kf ), and hence, rock’s stiff frameworkfeels the applied load at least as much as the pore fluid and deforms. Because the applied effective meanstress was significantly larger in the undrained compression, this specimen compacted twice as much as thedrained one (Figure 6b).

It is observed that in the undrained test, pore pressure is increasing from 2.8 to 7.2MPa when the sandstone iscompacting (Figure 7), and it starts decreasing only when the deformation of the specimen becomes inelastic(volume strain response deviates from linearity). The difference in peaks of pore pressure and volume straindiagrams in Figure 7 can be attributed to the initial microcracking of the specimen at the onset of inelasticity(γ~3×10�3). The microcracks start contributing to the increase in the pore volume; hence, the pore pressureincrements due to overall compaction of the specimen become smaller and eventually p starts decreasingat γ~6.5 × 10�3. However, the solid part keeps compacting at this point, which is still reflected in the smallpositive increments in volume strain until γ reaches 10.5 × 10�3 and volume strain of the specimen startsdecreasing. It also is noted that at the peak shear stress, the pore pressure was still larger than its initial value,because the undrained specimen compacted more than it dilated prior to failure.

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Figure 6. Volume strain and AE for the plane strain compressionexperiments: (a) the drained test started at P=5.1MPa and p=0.6MPa,and (b) the undrained test started at initial P= 10.0MPa and p= 2.8MPa.The undrained specimen compacted twice as much as the drained one,and in both tests, the rock dilated during inelastic deformation by asmaller amount than it compacted elastically.



4.2. Hardening Parameters

To compare the undrained behavior withthe constitutive model prediction, thehardening parameters such as dilatancyfactor β, friction coefficient μ, and effectivebulk modulus Keff need to be determinedfrom the corresponding drained test, wherethe effective mean stress was similar.4.2.1. Dilatancy FactorThe measured principal strains containboth elastic and plastic components. Theonset of the nonlinear response was usedas a benchmark signifying when plasticdeformation initiated (Figure 6a). Theincremental plastic strains were determinedby removing the calculated elastic responsefrom the measured deformation assumingno change in elastic parameters [Riedeland Labuz, 2007]:

Δεp ¼ Δεmeas � Δεe (27)Δγp ¼ Δγmeas � Δγe (28)

For the drained test conducted at 4.5MPa effective confinement, plastic deformation was only associatedwith a dilatant response (Figure 8). The increase of plastic volume strain was fairly linear from the onset ofinelasticity up to the peak. The dilatancy angle ψ [Hansen, 1958] was calculated from

sin ψ ¼ β ¼ � Δεp

Δγp(29)

and it is presented in Figure 8. It can be seen that for γp> 0.5 × 10�3, the dilatancy angle of the sandstonebecomes approximately constant and equal to 24°; hence, β = 0.41 (Figure 9). Note that dilatancy is meanstress dependent [Riedel and Labuz, 2007].4.2.2. Friction CoefficientThe Mohr-Coulomb linear yield function can be written as [Riedel and Labuz, 2007]

μ ¼ sinφ ¼ σ1 � σ3σ1 þ σ3 þ 2Vo � 2p (30)

where φ is the friction angle and Vo is the (theoretical) uniform triaxial tensile strength. The strengthparameter Vo is assumed to be constant during hardening and it was determined by performing a series of

conventional triaxial compression andextension tests and matching compressionand extension failure lines in the q-P′ plane[Meyer and Labuz, 2013]. The frictioncoefficient μ is presented as a function ofplastic shear strain γp (Figure 9), and μ isincreasing from 0.75 to 0.79 during theinelastic deformation process.4.2.3. Effective Bulk Modulus KeffKeff is the parameter that links theporoelastic response of the material withthe dilatant hardening model. To calculateKeff, the elastic change in the apparentfluid volume fraction Δυ= υ� υo duringdeformation needs to be determined. Fromequation (8), it can be seen that Δυ has

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peak

p

Figure 7. Volume strain ε and pore pressure pmeasured in an undrainedplane strain compression test. Pore pressure increased from 2.8 to7.0MPa when the specimen was compacting and then it decreased to4.4MPa at failure, which is larger than the initial pore pressure becausethe undrained specimen compactedmore than it dilated prior to failure.

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gle

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eg]

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vol

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stra

in

p[1

0-3 ]

Plastic shear strain p [10-3]

(-) p

Figure 8. Plastic volume strain εp and dilatancy angle ψ as functionsof plastic shear strain γp after the onset of inelastic deformation indrained plane strain compression.



“stress” in the numerator with order ofmegapascal and poroelastic bulk moduliin the denominator with order ofgigapascal. Hence, Δυ< 0.01 and is on theorder of the accuracy of the porositymeasurements, so it can be neglected andwe can assume that υ= υo=ϕ. Then, theequation for Keff can be simplified basedon poroelastic relationships [Detournayand Cheng, 1993] and becomes

Keff ¼ BKα (31)

Skempton coefficient B was observed tobe changing with applied pore pressuregiven the effective mean stress being

preserved constant (Figure 10). It was found that the full saturation of the rock specimens at P′= 5.0MPa wasachieved when the pore pressure exceeded 4MPa [R. Y. Makhnenko and J. F. Labuz, 2013]. Then, B becameconstant and after making the corrections for extra water volume in the drainage system (equation (23))and the application of deviatoric loading, B=0.61 [Makhnenko, 2013].

The other poroelastic parameters measured in drained and unjacketed experiments performed at P′=5.0MPa[R. M. Makhnenko and J. F. Labuz, 2013] are the following:

K ¼ 9:5GPa; K ′s ¼ 30GPa; α ¼ 0:70

Substituting these values into (31), we obtain Keff = 8.3 GPa. Shear modulus of Berea sandstone at P′=5MPa iscalculated from the initial part of loading in Figure 6a:

G ¼ ΔτΔγe

¼ 4:6GPa (32)

Because mean stress is changing during plane strain compression, the inelastic hardening modulus H isdetermined from

H ¼ Δτ � μΔPΔγ� ΔτG

(33)

However, based on the volume strain data (Figures 6 and 7), it can be observed that the changing mean stresscauses the sandstone specimens to compact during elastic deformation by a larger amount than they dilate

when the response becomes inelastic, and thismakes dilatant hardening more challenging tostudy. Thus, drained and undrained tests, where themean stress can be preserved constant, provide theopportunity to directly observe dilatant hardening.

4.3. Compression With Constant Mean Stress

Cylindrical specimens were tested in conventionaltriaxial compression under drained and undrainedconditions (Figures 11a and 11b), whereP= (σa+ 2σr)/3 = const can be preserved byapplying increments of Δσa> 0 (axial strainrate ~10�6 s�1) and decreasing the radial stress(cell pressure) by Δσr=�Δσa/2< 0. A drainedconventional triaxial experiment was performedunder the conditions of P= const = 20.0MPaand p= const = 3.0MPa. The initial mean stress wassignificantly larger than the one used in the plane

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cy f

acto

r

Fri

ctio

n c

oef

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ent

Plastic shear strain p [10-3]

Figure 9. Development of the friction coefficient μ and dilatancy factor βas a function of plastic shear strain γ p in drained plane strain compression.

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mp

ton

co

effi

cien

t B

Pore pressure p [MPa]

B max = 0.61 (at P = 5 MPa)

B max = 0.40 (at P = 17 MPa)

Figure 10. Change in Skempton coefficient Β with increase inpore pressure p measured at P′= 5.0MPa (red dots) and P′= 17.0MPa (blue dots). At P′= 5.0MPa, the sandstone specimenwas considered to be saturated at pore pressure p> 4MPa,where maximum Bwas reached: Bmax = 0.61. For the specimentested at P′=17.0MPa, Bmax = 0.40 was achieved at p> 3.5MPa.



strain compression test reported in theprevious section because the failure of thespecimen for the case of constant meanstress loading can be assured only ifP′= P� p>UCS/3≈ 15MPa is used. Thespecimen failed at σ1 =51.0MPa andσ2 = σ3 = 4.5MPa. The results of the drainedtest in terms of volume strain, AE activity,and pore pressure are presented inFigure 11a.

As shown in Figure 11a, monotonicallydecreasing volume strain indicates thatthe drained specimen started dilatingfrom the beginning of the deviatoricloading. It is also confirmed by the AEactivity and the observations from theinitial loading-unloading procedure, whichindicated the presence of permanentstrain. Note that for the case of constantP, there is no elastic volume change forisotropic response: Δεe=0 and Δεp=Δεmeas

(measured volume strain). An incrementof shear strain consists of elastic and plasticparts; hence, Δγmeas =Δγe+Δγp=Δτ/G+Δγp. Then, the friction coefficient μ(Figure 12) can be calculated for this loadingcondition using the method described insection 4.2.2.

The friction coefficient was found to bemonotonically increasing, being equal to0.72 at the peak stress, which is smallerthan μ measured in the plane straincompression experiment possibly due to

an intermediate stress effect. The dilatancy factor β was obtained from the plane strain test reported inFigure 9 and β = 0.41 for P′= 17.0MPa, which is consistent with other tests.

The effective bulk modulus Keff was also calculatedfor the mean stress P=20.0MPa and p= 3.0MPa.The specimen was found to be fully saturated atP′=17.0MPa when the back pressure exceeded3.0MPa and the maximum Skempton coefficientwas Bmax = 0.40 (Figure 10). Drained bulk moduluswas measured to be K=11.3GPa and shear modulusG=6.4GPa. The unjacketed bulk modulus wasfound to be mean stress independent at the rangeof pressures applied in the tests [R. M. Makhnenkoand J. F. Labuz, 2013], Ks′=30GPa and α=0.62. Usingequation (31), Keff = 7.3GPa is obtained. Table 1 is asummary of the material parameters.

To evaluate the quality of the model prediction, theundrained conventional triaxial test was performedwith the same initial conditions as the drained test:P= const = 20.0MPa and po=3.0MPa. The mass of

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v

p = 3 MPa

p = 3 MPa

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a)

b)

Figure 11. Volume strain εv (dark blue), pore pressure p (red), and AEactivity (grey) as functions of shear strain εs in conventional triaxialcompression experiments conducted at constantmean stress P=20.0MPaand initial pore pressure po= 3.0MPa under (a) drained conditions,where p= const = 3.0MPa, and (b) undrained conditions, where porepressure was changing from 3.0 to 0.5MPa.

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Fri

ctio

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t

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Figure 12. Development of friction coefficient μ in theconventional triaxial compression test conducted underdrained conditions. The mean stress P and pore pressure pwere preserved constant throughout the test: P = 20.0 MPaand p = 3.0 MPa.



the fluid inside the specimen was kept constant, so the pore pressure was allowed to change. During theundrained test, p decreased from 3.0 to 0.5MPa, which produced the increase in effective confinement anddelayed the specimen failure to the point where σ1 = 55.2MPa and σ2 = σ3 = 2.4MPa. The development of thevolume strain, AE activity, and pore pressure with shear strain in undrained conventional triaxial compressionis presented in Figure 11b.

In the undrained test conducted at constant mean stress, no initial compaction was observed and thespecimen dilated by approximately the same amount as the one tested under the drained condition. AEactivity was observed from the beginning of deviatoric loading, which confirmed the inelastic nature of theundrained deformation.

Moreover, an undrained plane strain experiment was conducted preserving P=const =20.0MPa with the sameinitial pore pressure po=3.0MPa. Using equation (13) and neglecting the deformation of the prismatic specimenin the intermediate principal stress direction (ε2=0), the following relationship can be written for the mean stress:

P ¼ σ1 þ σ2 þ σ3ð Þ=3 ¼ 1þ νuð Þ σ1 þ σ3ð Þ=3 (34)

Therefore, when applying axial load to the specimen Δσ1> 0, σ3 was preserved as

σ3 ¼ 3P= 1þ νuð Þ � σ1 (35)and νu was measured to be 0.35. The undrained plane strain compression test was stopped at σ1 = 43.6MPa,σ2 = 15.6MPa, σ3 = 0.8MPa, and p= 0.7MPa. The specimen did not fail at this point, but the condition ofσ3′= σ3� p ≤ 0 was avoided. The results of this test in terms of volume strain, AE activity, and pore pressureare presented in Figure 13.

Volume strain and pore pressure response in the plane strain test are observed to be similar to the conventionaltriaxial test, even though the change in their values is smaller; this happened because the plane strain test wasstopped at a smaller shear strain value compared to the conventional triaxial test. Results of the drained and twoundrained experiments conducted at constant P and the constitutivemodel prediction are presented in Figure 14.

It is noted that up to a certain point, the axisymmetric and plane strain loading give similar results in termsof undrained behavior. Also, unlike in the plane strain tests conducted at constant confining pressure, the

initial undrained response observed inthe constant mean stress experimentsis stiffer than the drained one; this isdue to the dilatant hardening effectfrom the beginning of shear loading.The constitutive model predicts theundrained behavior of the fluid-saturatedsandstone up to approximately 95% ofthe peak shear stress. After this point,each increment of shear strain requiresa much larger increment in shear stresscompared to the beginning of loadingand other processes may play a role. Forexample, some heating can be associatedwith the shearing of fluid-saturated rockand if the thermal effect is considered,as shown by Sulem et al. [2011], thenit decreases the effect of dilatanthardening in the undrained compression.This behavior is not captured by the

Table 1. Material Parameters of Berea Sandstone Used in Dilatant Hardening Model

Test Conditions K (GPa) α B Keff (GPa) G (GPa) μ

Plane strain σ3 = const = 5.0 MPa 9.5 0.70 0.61 8.3 4.6 0.75–0.79Conventional triaxial P = const = 20MPa 11.3 0.62 0.40 7.3 6.4 0–0.72

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ts

Shear strain [10-3]

po = 3 MPa

p

AE

Figure 13. Volume strain ε (dark blue), pore pressure p (red), and AE activity(grey) plotted as functions of shear strain γ for an undrained plane straincompression experiment conducted at constant mean stress P = 20.0MPaand pore pressure p changing from 3.0MPa at the beginning of the test to0.7MPa at the point when the test was terminated.



model, which still can be treated as a first-orderapproximation to the inelastic behavior of fluid-saturated porous rock. Furthermore, homogeneityof deformation is assumed, and even thoughno localization was observed from the lateraldisplacements, some clustering of AE events wasdetected prior to peak stress in dry tests [Iversonet al., 2007; Riedel and Labuz, 2007]. Indeed, theeffect of dilatant hardening on the developmentof concentrated strain in a narrow zone that isslightly weaker than the surrounding rock couldexplain this behavior [Rudnicki, 1985; Rudnickiand Chen, 1988].

5. Conclusions

The University of Minnesota plane strain apparatuswas used for testing poroelastic and inelasticresponse of water-saturated Berea sandstone.Prismatic specimens wedged in a stiff frame weresubjected to approximate plane strain conditions,meaning that the deformation in the intermediate

principal stress direction was negligible. The apparatus provides the ability to accurately determine volume andshear strains by measuring the displacements in the plane where major and minor principal stresses act.Additionally, conventional triaxial tests were performed on cylindrical specimens and axial and tangentialstrains weremeasured. By testing the sandstone at the limiting conditions of drained, undrained, and unjacketed,the poroelastic parameters of the rock, such as drained and unjacketed bulk moduli and Skempton B coefficient,were determined at two different values of effective mean stress (5.0 and 17.0MPa).

The results of drained and undrained plane strain and conventional triaxial compression experiments werediscussed in the framework of a dilatant hardening constitutive model. The parameters that govern theinelastic deformation of fluid-saturated rock, i.e., dilatancy factor β, friction coefficient μ, poroelasticcoefficient Keff, shear modulus G, and inelastic hardening modulus H, were calculated from the drainedresponse. The onset of inelastic response was determined from deviation of volume strain response fromlinearity, which coincided with an increase in acoustic emission (AE) activity. Dilatancy and friction werefound to be pressure dependent, and the sandstone behavior was nonassociative: friction and dilatancyangles were different. When the mean stress was preserved constant during loading, then the modelprediction based on the drained compression tests was found to be consistent with the results of twoundrained tests, one conventional triaxial and one plane strain compression.

Knowledge of dilatant hardening behavior is essential for characterization of fluid-saturated rock and theproper assessment of underground structures, such as long-term CO2 storage facilities. The difficulty andexpense in making relevant field observations ensure that laboratory measurements will continue tocontribute to understanding coupled deformation-diffusion effects.

ReferencesBernabé, Y., and W. F. Brace (1990), Deformation and fracture of Berea sandstone, in The Brittle-Ductile Transition in Rocks, The Heard Volume,

Geophys. Monogr. Ser., vol. 56, edited by A. G. Duba et al., pp. 91–101, AGU, Washington, D. C.Biot, M. A. (1941), General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155–164.Biot, M. A. (1973), Nonlinear and semilinear rheology of porous solids, J. Geophys. Res., 78, 4924–4937, doi:10.1029/JB078i023p04924.Bishop, A. W. (1976), Influence of system compressibility on observed pore pressure response to an undrained change in stress in saturated

rock, Géotechnique, 26(2), 371–375.Brace, W. F., and R. J. Martin (1968), A test of the law of effective stress for crystalline rocks of low porosity, Int. J. Rock Mech. Min. Sci., 5, 415–436.Detournay, E., and A. Cheng (1993), Fundamentals of poroelasticity, chap. 5, in Comprehensive Rock Engineering: Principles, Practice and

Projects, Vol. II, Analysis and Design Methods, edited by C. Fairhurst, pp. 113–171, Pergamon Press, Oxford, U. K.Duda, M., and J. Renner (2013), The weakening effect of water on the brittle failure strength of sandstone, Geophys. J. Int., 192(3), 1091–1108.Han, C., and I. G. Vardoulakis (1991), Plane-strain compression experiments on water-saturated fine-grained sand, Géotechnique, 41(1), 49–78.Hansen, B. (1958), Line ruptures regarded as narrow rupture zones: Basic equations based on kinematic considerations, in Proceedings of

Conference Earth Pressure Problems, Brussels, Belgium 1, pp. 39–49.

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Shear strain [10-3]

Sh

ear

stre

ss [

MP

a]

undrained

Triaxial, undrained

Plane strain,

Triaxial, drained

Model, undrained

Figure 14. Results of the compression experiments, drained(grey) and undrained (dark blue) conventional triaxial andundrained plane strain (blue). The prediction of the elasto-plasticmodel for the undrained response is shown in red and is in agood agreement with the experimental data up to about 95% ofthe peak shear stress. Shear stress is taken as τ (= q/2 in caseof conventional triaxial experiments) and shear strain is γ incase of plane strain test, and 2εs for the conventional triaxialexperiments (to be conjugate with τ).

AcknowledgmentsThe data are available through theUniversity of Minnesota Conservancy bycontacting the corresponding author.Partial support was provided by DOEgrant DE-FE0002020 founded throughthe American Recovery and ReinvestmentAct. J. Meyer assisted in developing theexperimental apparatus.



http://dx.doi.org/10.1029/JB078i023p04924

Iverson, N., C.-S. Kao, and J. F. Labuz (2007), Clustering analysis of AE in rock, J. Acoustic Emission, 25, 364–372.Labuz, J. F., S.-T. Dai, and E. Papamichos (1996), Plane-strain compression of rock-like materials, Int. J. Rock Mech. Min. Sci., 33(6), 573–584.Lei, X., T. Tamagawa, K. Tezuka, and M. Takahashi (2011), Role of drainage conditions in deformation and fracture of porous rocks under

triaxial compression in the laboratory, Geophys. Res. Lett., 38, L2430, doi:10.1029/2011GL049888.Makhnenko, R., and J. Labuz (2014), Plane strain testing with passive restraint, Rock Mech. Rock Eng., 47(6), 2021–2029, doi:10.1007/s00603-

013-0508-2.Makhnenko, R. M., and J. F. Labuz (2013), Unjacketed bulk compressibility of sandstone in laboratory experiments, Poromechanics V—Proceedings

of the 5th Biot Conference on Poromechanics, pp. 481–488, doi:10.1061/9780784412992.057.Makhnenko, R. Y. (2013), Deformation of fluid-saturated porous rock, PhD thesis, Dep. of Civil Eng., Univ. of Minn., Minneapolis.Makhnenko, R. Y., and J. F. Labuz (2013), Saturation of porous rock and measurement of the B coefficient, paper 468 presented at 47th U.S.

Rock Mechanics/Geomechanics Symposium, San Francisco, Calif., 23–26 June.Makhnenko, R. Y., C. Ge, and J. F. Labuz (2012), AE from undrained and unjacketed tests on sandstone, paper 581 presented at 46th U.S. Rock

Mechanics/Geomechanics Symposium, Chicago, Ill., 24–27 June.Meyer, J. P., and J. F. Labuz (2013), Linear failure criteria with three principal stresses, Int. J. Rock Mech. Min. Sci., 60, 180–187.Mokni, M., and J. Desrues (1998), Strain localization in undrained plane-strain biaxial tests on Hostun RF sand, Mech. Cohes. Frict. Mater., 4,

419–441.Nur, A. (1972), Dilatancy, pore fluids, and premonitory variations of ts /tp travel times, Bull. Seismol. Soc. Am., 62, 1217–1222.Oye, V., E. Aker, T. M. Daley, D. Kühn, B. Bohloli, and V. Korneev (2013), Microseismic monitoring and interpretation of injection data from the

In Salah CO2 storage site (Krechba), Algeria, Energy Procedia, 37, 4191–4198.Renner, J., B. Evans, and G. Hirth (2000), On the rheologically critical melt fraction, Earth Planet. Sci. Lett., 181, 585–594.Rice, J. R. (1975), On the stability of dilatant hardening for saturated rock masses, J. Geophys. Res., 80(11), 1531–1536, doi:10.1029/

JB080i011p01531.Rice, J. R. (1977), Pore pressure effects in inelastic constitutive formulations for fissured rock masses, in Advances in Civil Engineering Through

Engineering Mechanics, pp. 360–363, Am. Soc. Civ. Eng., New York.Rice, J. R., and M. P. Cleary (1976), Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible

constituents, Rev. Geophys. Space Phys., 14, 227–241.Riedel, J. J., and J. F. Labuz (2007), Propagation of shear band in sandstone, Int. J. Numer. Anal. Methods Geomech., 31, 1281–1299.Rudnicki, J. W. (1985), Effect of pore fluid diffusion on deformation and failure of rock, in Mechanics of Geomaterials, edited by Z. P. Bažant,

pp. 315–347, John Wiley, New York.Rudnicki, J. W., and C.-H. Chen (1988), Stabilization of rapid frictional slip on a weakening fault by dilatant hardening, J. Geophys. Res., 93(B5),

4745–4757, doi:10.1029/JB093iB05p04745.Schmitt, D. R., and M. D. Zoback (1992), Diminished pore pressure in low-porosity crystalline rock under tensional failure: Apparent

strengthening by dilatancy, J. Geophys. Res., 97(B1), 273–288, doi:10.1029/91JB02256.Scholz, C. H. (1968), Microfracturing and the inelastic deformation of rock in compression, J. Geophys. Res., 73(4), 1417–1432, doi:10.1029/

JB073i004p01417.Sulem, J., and H. Ouffroukh (2006), Hydromechanical behavior of Fontainebleau sandstone, Rock Mech. Rock Eng., 39(3), 185–213.Sulem, J., I. Stefanou, and E. Veveakis (2011), Stability analysis of undrained adiabatic shearing of a rock layer with Cosserat microstructure,

Granular Matter, 13, 261–268.Terzaghi, K. (1943), Theoretical Soil Mechanics, 510 pp., John Wiley, New York.Viesca, R. C., E. L. Templeton, and J. R. Rice (2008), Off-fault plasticity and earthquake rupture dynamics: 2. Effects of fluid saturation,

J. Geophys. Res., 113, B09307, doi:10.1029/2007JB005530.Yale, D. P., T. Mayer, and J. Wang (2010), Geomechanics of oil sands under injection, paper 257 presented at 44th U.S. Rock Mechanics/

Geomechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, Salt Lake City, Utah, 27–30 June.



http://dx.doi.org/10.1029/2011GL049888http://dx.doi.org/10.1007/s00603-013-0508-2http://dx.doi.org/10.1007/s00603-013-0508-2http://dx.doi.org/10.1061/9780784412992.057http://dx.doi.org/10.1029/JB080i011p01531http://dx.doi.org/10.1029/JB080i011p01531http://dx.doi.org/10.1029/JB093iB05p04745http://dx.doi.org/10.1029/91JB02256http://dx.doi.org/10.1029/JB073i004p01417http://dx.doi.org/10.1029/JB073i004p01417http://dx.doi.org/10.1029/2007JB005530

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