The transition from brittle faulting to cataclastic flow in porous sandstones' Mechanical deformation

Transcription

1 JOLrRNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B2, PAGES , FEBRUARY 10, 1997 The transition from brittle faulting to cataclastic flow in porous sandstones' Mechanical deformation Teng-fong Wong, Christian David, and Wenlu Zhu Department of Earth and Space Sciences, State University of New York at Stony Brook Abstract. Triaxial compression experiments were conducted to investigate the inelastic and failure behavior of six sandstones with porosities ranging from 15% to 35%. A broad range of effective pressures was used so thathe transition in failure mode from brittle faulting to cataclastic flow could be observed. In the brittle faulting regime, shear-induced dilation initiates in the prepeak stage at a stress level C' which increases with effective mean stress. Under elevated effective pressures, a sample fails by cataclastic flow. Strain hardening and shear-enhanced compaction initiates a stress level C* which decreases with increasing effective mean stress. The critical stresses C' and C* were marked by surges in acoustic emission. In the stresspace, C* maps out an approximately elliptical yield envelope, in accordance with the critical state and cap models. Using plasticity theory, the flow rule associated with this yield envelope was used to predict porosity changes which are comparable to experimental data. In the brittle faulting regime the associated flow rule predicts dilatancy to increase with decreasing effective pressure in qualitative agreement with the experimental observations. The data were also compared with prediction of a nonassociative model on the onset of shear localization. Experimental data suggest that a quantitative measure of brittleness is provided by the grain crushing pressure (which decreases with increasing porosity and grain size). Geologic data on tectonic faulting in siliciclasfic formations (of different porosity and grain size) are consistent with the laboratory observations. Introduction Jamison and Stearns, 1982; Antonellini et ai., 1994] and fluid flow [e.g., Hippler, 1993; Atonellini and Aydin, 1994] in When subjected to an overall compressive loading, a rock sandstone formations, as well as geotechnical problems including often fails by shear localization or by cataclastic flow. Under reservoir compaction [e.g., Teufel et al., 1991], borehole stability elevated temperatures and pressures, homogeneous plastic flow [e.g., Veeken et al., 1989], and drilling [e.g., Suarez-Rivera et al., can also develop through thermally activated dislocation activity 1990]. In their seminal study, Handin et al. [1963] have and diffusive mass transfer [Paterson, 1978]. Cataclastic flow underscored the dominant role of effective pressure in controlling represents the key intermediate step in the brittle-plastic the transition from brittle faulting to cataclastic flow in porous transition in quartzo-feldspathic aggregates [Tullis and Yund, sedimentary rocks. Subsequent researchas shown that since the 1992; Hirth and Tullis, 1994]. Recent studies of the porosity evolution of porosity and stres state are coupled, porosity also change and yield stress in crustal rocks of different porosities exerts important control over failure mode [Logan, 1987; Wong, have demonstrated that there are two fhndamenta!ly different 1990; Scott and Nielson, 1991]. During cataclastic flow, types of cataclastic flow. In low-porosity rocks, dilation of the significant shear-enhanced compaction can be induced by the pore space is commonly observed, and the yield stresses for this application of a deviatoric stress [Schock et al., 1973; Zhang et type of dilatant catac!astic flow show a positive pressure al., 1990a]. The reduction in porosity allows the rock to work dependence [Fredrich et al., 1989; Fischer and Paterson, 1989]. harden, thus inhibiting the development of shear localization. In contrast, appreciable porosity decreases are generally observed Alternatively, the deviatoric stress can cause dilatancy, which in high-porosity rocks [Edmond and Paterson, 1972; Jamison may ultimately lead to failure by shear localization and brittle and TeuJ l, 1979], and the yield stress for this type of compactive faulting under relatively low effective pressure [Dunn et al., cataclastic flow decreases with increasing effective pressure 1973; Bernabd and Brace, 1990]. [I ong et al., 1992]. Although the two modes of failure are very different on the This study focuses on the transition from brittle faulting to macroscopic scale, they both involve micromechanical processes compactire cataclastic flow in porousandstones. The laboratory which are thought to be brittle since pervasive grain-scale investigations provide important physical insights into tectonic microcracking and acoustic emission activity [Lockner et al., processes in relation to faulting [e.g., Aydin and Johnson, 1978; 1992; Wong et al., 1992; Read et al., 1995] are commonly observed. Coalescence of the stress-induced microcracks leads to shear localization, whereas cataclastic flow is a manifestation of Now at in- titut de Physique du Globe, Strasbourg, France. homogeneously distributed microcracking [Mendndez et ai., Copyright 1997 by the American Geophysical Union. The rock which has been most thoroughly studied in the Paper number 96JB laboratory is Berea sandstone. Although sandstones of different 01484)227/97/96JB porosity, grain size, cementation and clay content have also been 3009

2 3010 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES were from the same blocks (kindly furnished by D. Jeannette) considered, our understanding of the phenomenology and micromechanics of brittle faulting and cataclastic flow in these previously studied by David et al. [1994]. The Darley Dale otherocks is not as comprehensive. Therefore the first objective sandstone (kindly furnished by P. G. Meredith) is considered to of the present study is to acquire a relatively complete set of mechanical data (including stress, strain, porosity change, and be similar to the samples recently studied by Read et al. [1995]. The Boise sandstone (kindly furnished by J. T. Fredrich)has acoustic emission activity) for six sandstones with porosities porosity comparable to the samplestudied by Zhang et al. r"anging from!5% to 35%. For hydrostaticompaction a [1990b] but has a significantly smaller grain size. To differentiate previoustudy of Zhang et al. [1990b] has shown that there the two blocks, we will refer to the former and the latter as Boise exists a critical stresstate (corresponding to the critical effective II and Boise I, respectively. The Berea, Boise, Darley Dale, and pressure for the onset of grain crushing) which is primarily Kayenta samples were from cores perpendicular to bedding, and controlled by the porosity and grain size. Under nonhydrostatic the Adamswilier samples were from cores parallel to bedding. loading the failure and compactive yield are characterized by Two suites of Rothbach samples parallel and perpendicular to several critical stress parameters, including the stress at the onset bedding were investigated, and they will be referred to as of dilatancy, the peak stress for brittle fracture, the yield stress at the onset of shear-enhanced compaction, and the effective pressure at the transition from brittle faulting to cataclastic flow. To what extent are these critical stress states controlled by the Rothbach 1 and Rothbach 2, respectively. The sandstoneselected have relatively small variations in porosity within the same block. The (interconnected) porosity of a sample was determined from the weight difference between the microstructural attributes? Shear-induced dilation and dried and saturated core. The grain size was determined by compaction are observed in the brittle fracture and cataclastic measuring the mean intercept length L. of solid grains under an flow regimes, respectively. How are these inelastic strains related optical microscope. Since the data will be interpreted using an to the critical stresses? Plasticity theory [e.g., Desai and elasticontact model with impinging spheres, we also) need to Siriwardane, 1984] and bifurcation analysis [e.g., Rudnicki and estimate the equivalent grain radius Rg. From geometric Rice, 1975] have been used to predict the inelastic porosity probability [Underwood, 1970], it can be shown that if the grains change and onset of shear localization, respectively. To what extent do the experimental observations agree with the theoretical predictions? The inelastic deformation results in significant changes in the pore geometry, which can in turn are assumed to be spherical, then the mean intercept length is related to grain radius by Rg = (3/4) L3. The grain sizes for Berea and Kayenta sandstones were reported by Zhang et al. [!990b], and grain sizes for Adamswiller and Rothbach were reported by influence the micromechanics of failure as well as fluid transport David et al. [1994]. In the latter paper the grain radii of properties. How does the hydraulic permeability evolve with failure in the brittle faulting and compactive cataclastic flow regimes? To investigate these questions, we conducted a parallel series of experiments on mechanical deformation and permeability evolution, which are summarized in this paper and a companion paper [Zhu and Wong, this issue], respectively. Adamswilier and Rothbach sandstones were calculated by the wrong formula (Rg = L. /2); this mistake has been corrected in Table 1. Grain sizes for the Darley Dale and Boise II sandstones were determined by us. It should be noted that we did not include the contributions of clay and mica. It is difficult to resolve the former under the optical microscope, and the latter has elongate geometry very different from that of a sphere. We believe that Mechanical Data and Critical Stress States our estimates of the effective grain radii are relevanto the analysis of grain fracture processes which are dominated by Sandstones Studied The sand}tones were selected because of their wide range of porosity and grain size. Petrophysical description of the rocks is provided in Table 1. All the Kayenta sandstone data and some of the Berea sandstone data were previously reported by Wong et al. [!992]. The Kayenta and Berea sandstone samples were from the microcracks initiating from contacts between the quartz and feldspar grains. Experimental Procedure The jacketed samples were saturated with distilled water and stressed in the conventional triaxial configuration at room same blocks studied by Zhang et al. [1990b]. The Adamswilier temperature. Kerosene was used as the confining medium. Both and Rothbach sandstones (from the Vosges mountains, France) confining pressure and pore pressure were monitored by strain- Table 1. Petrophysical Description of Six Sandstones Investigated in This Study Porosity Mean Intercept Grain Sandstone O, % Length L3, mm Radius Rg, mm Modal Analysis Adamswilier quartz 71%, feldspar 9%, oxides and mica 5%, clay -11% Berea quartz 71%, feldspar 10%, carbonate 5%, clay ~10% Boise II quartz 67%, feldspar 14%, mica 2%, clay 13% Darley Dale quartz 66%, feldspar 21%, mica 3%, clay 6% Kayenta quartz 81%, feldspar 16%, carbonate 1% Rothbach quartz 68%, feldspar 16%, oxides and mica 3%, clay

3 WONG ET AL.: MECHANICAL DEFORMATION 1N POROUS SANDSTONES 3011 gage pressure transducers to accuracies of 0.5 and MPa, respectively. All th experiments were conducted at a fixed pore pressure of 10 MPa, with confining pressures ranging from 13 to 550 MPa. Adjustment of a pressure generator kept the pore pressure constant, and the pore volume change was recorded by monitoring the piston displacement of the pressure generator with a displacement transducer (DCDT). The porosity change was calculated from the ratio of the pore volume change to the initial bulk volume of the sample with an uncertainty of +_0.1%. The axial load was measured with an external load cell with an accuracy of 1 kn. The displacement was measured outside the pressure vessel with a DCDT mounted between the moving piston and the fixed upper platen. The uncertainty of the axial displacement measurement was 10 gm. Except for the Darley Dale sandstone samples (which were of diameter 20 mm and length 40.6 mm), the cylindrical samples ha diameters of 18.4 mm and length 38.1 mm and were jacketed with polyolefine tubing. The axial displacement was servo- controlled at a fixed rate (corresponding to a nominal strain rate of 5 x 10'5/s) which was sufficiently slow to ensure fully "drained" deformation. With a knowledge of the stiffness of the 4 ""'-- loading system (2.38 x!0 N/m), the axial displacement of the sample was obtained by subtracting the displacement of the o 6 "--,.,,.. loading system from the apparent displacement recorded by the DCDT, and the axial strain was calculated with reference to the initial length of the sample. To calculate the axial stress from the recorded axial force, effect of bulging was accounted for by lo inferring the relative increase in area of a deformed sample by Figure 1. Mechanical data for Adamswilier sandstone. subtracting the axial strain from the porosity change. Differential stress and porosity decrease were plotted versus axial To measure acoustic emission (AE) activity during the triaxial strain. Effective pressures were as indicated. The solid curves are experiments, we used a piezoelectric transducer (PZT-7, 5.0 mm diameter, 1 MHz longitudinal resonant frequency) on the flat for samples which failed by shear localization, and the dashed curves are for samples which failed by cataclastic flow. surface of a steel spacer attached to the jacketed sample. The AE signals were conditioned by a preamplifier (gain 40 db, frequency response 1.5 khz- 5 MHz). To distinguish AE events from electric spikes, a discriminator was used to check two porosity initially decreased, but near the peak stress it reversed to characteristics of the incoming signal. The details were described an increase indicating dilation of the pore space. The dilation by Zhang et al. [1990a]. decreased with increasing effective pressure. Visual inspection of Mechanical Data: Stress, Strain and Porosity Change We will adopthe convention that compressive stresses and compactire strains (i.e., shortening and porosity decrease) are postpeak samples confirmed that they failed by shear localization, with a thoroughgoing shear band cutting across each sample. The dashed curves (for samples deformed at effective positive, and we will denote the maximum and minimum pressures of 60, 100, and 150 MPa) are representative of the (compressire) principal stresses by ( and ( 3, respectively. The compactive cataclastic flow regime. The slopes of the differential pore pressure will be denoted by P/, and the difference between stress-axial strain curve were nonnegative, and the porosity the confining pressure (Pc = c 2 = (x3) and pore pressure will be decreased monotonically with deformation. Shear localization referred to as the "effective pressure" Pelf The complete set of was not evident in samples deformed to an axial strain of up to mechanical data for Adamswilier sandstone are shown in Figure 30%. The sample deformed at 40 MPa showed a peak stress and 1. The top graph shows the differential stress c,- ( 3 versus the strain softening, but the porosity persistently decreased with axial strain for six triaxial compression experiments at a fixed deformation. This "transitional" mode of failure is manifested by pore pressure of 10 MPa and with confining pressures the development of several conjugate shear bands in the sample. maintained at! 5, 30, 50, 70, 110 and 160 MPa, respectively. The Additional insights are gained by plotting the effective mean bottom graph shows porosity decrease versus axial strain for the stress ((x, + 2%)/3 - Pp versus the porosity change (Figures 2a same samples. and 2b). Previou studies have demonstrated that the hydrostatic The samples deformed at an effective pressures of 5 MPa and and nonhydrostatic (shear) stresses have fundamentally different 20 MP are representative of the brittle faulting regime. The effects on the porosity. A hydrostatic stress always induces differential stress attained a peak, beyond which strain softening porosity decrease. For a porous sandstone, the typical hydrostat was observed and the stress progressively dropped to a residual (as indicated by the dashed curves in Figure 2a and 2b) shows an level. The loading frame wasufficiently stiff that the stress drop inflection point which marks the critical effective pressure for the process wa stable in every experiment. The peak stress shows a onset of grain crushing and pore collapse, as indicated by positive correlation with effective pressure, which is typical of microstructural and AE measurements [Zhang et al., 1990b, c]. This critical effective pressure will be denoted by P*. Mohr-Coulomb type of brittle failure [Paterson, 1978]. The 270[ oo ,.. oo? a 150[' 180 [ 60MPa { 6o O,,a 30 _ axial s[ain (%). 0 I 5MPa O... '... OMPa _ ' "" '...,,,q:::,,, MPa

4 WONG ET AL.' MECHANICAL DEFORMATION IN POROUSANDSTONES 125 ; 100 ', E 5o 40MPa as C' (Figure 2a). This implies that at stress levels beyond C' the deviatoric stress field induced the pore space to dilate. This phenomenon is commonly observed in the brittle fracture regime, and we will refer to it as "shear-induced dilation". This behavior is akin to the dilatancy phenomenon commonly observed in lowporosity crystalline rocks [Brace, 1978]. As discussed above, a transitional type of failure was observed at the effective pressure of 40 MPa. Before the peak stress was attained, the porosity increased somewhat relative to the hydrostat (Figure 2a), buthen there was an accelerated decrease in porosity during the postpeak stage. o 25 Acoustic Emission Activity and Micromechanics of Failure (b) porosity change (%) Representative data for AE activity of the Rothbach I sandstone are shown in Figures 3 and 4. The cumulative AE count (dotted curve) and effective mean stress (solid curve) are plotted as functions of the porosity change for a sample deformed in the brittle faulting (Figure 3 a) and cataclastic flow (Figure 3b) regimes, respectively. For reference the hydrostat (dashed curve) is also included in the graphs. Intense AE activity was observed in both the brittle faulting and cataclastic flow regime. The onset of shear-inducedilation (C') and shear-enhanced compaction (C*) were both marked by surges in AE activity. In parallel with " 150MPa porosity change (%) Figure 2. (a) Shear-induced dilation and (b) compaction in Adamswilier sandstone. The solid curves show the effective mean stress as a function of porosity change for triaxial compression tests at fixed effective pressures as indicated. Failure modes are indicated. For reference the hydrostat is shown as dashed curves. The critical stress states C' and C* are indicated by the arrows. The nonhydrostatic and hydrostatic loadings are coupled together in a triaxial compression experiment. If the porosity change is solely controlled by the hydrostatic stresses, then the triaxial data (solid curves) should coincide with the hydrostat (dashed curves) in Figure 2. Deviations from the hydrostat would imply that additional porosity change was induced by the o 2.0E5 deviatoric stresses. In the cataclastic flow regime, the triaxial curve for a given effective pressure coincided with the hydrostat 5o up to a critical stresstate (indicated by C* in Figure 2b), beyond 0 which there was an accelerated decrease in porosity in comparison to the hydrostat. At stress levels beyond C* the porosity change (%) deviatoric stress field provided significant contribution to the Figure 3. Cumulative acoustic emission (dotted compactive strain, and this phenomenon is referred to as "shear- effective mean stress (solid curve) as 'functions enhanced compaction" [Curran and Carroll, 1979, 'ong et al, change in Rothbach sandstone. The samples failed by 1992]. fracture at effective pressure of 20 MPa, Rothbach 1, and In contrast, the porosity change behavior at effective pressures cataclastic flow at effective pressure of 140 MPa, of 5 and 20 MPa was such that the compaction decelerated in Foreference the hydrostat is shown as the dashed comparison to the hydrostat beyond critical stresstates marked critical stres states C' and C* are indicated by the arrows ' 30 E ' '"-','"' '1',' " ",,'1'' '1"''' '*'' I, 0, ,2 1.6 porosity change (%) E4 1.5E4 LU 1.2E4 < 9.0E3 ' E 6.0E3:3 o 3.0E3 O.OEO.._..250,(b),,' 4.0E5 / E 1.0E5 o: O.OEO curve)and of porosity (a) brittle (b) Rothbach 2. curves, The

5 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES 3013 f O O O axial strain (%) ' so 8 12 axial strain (%) 3OO Figure 4. Acoustic emission rate (dotted curve) and differential stress (solid curve) as functions of axial strain in Rothbach deformation parameters to emphasize the features common to all sandstone. The samples are the same as those shown in Figure 3 the experimental data and to provide a basis for quantitative which failed by (a) brittle fracture at effective pressure of 20 comparison. MPa, Rothbach 1 and (b) cataclastic flow at effective pressure of In the brittle regime we focus on the stress at the onset of 140 MPa, Rothbach 2. The critical stress states C' and C* are shear-inducedilation C' and the peak stress. Table 2 includes indicated by the arrows. all our samples for which these two critical stresses can be clearly determined. In Figure 5 the compiled data are shown in the stress space, with coordinates given by the effective mean stress P ( = the porosity measurements, the AE data were also used to map ({s + 2(s.0/3 - Pp) and the differential stress Q ( = (r - (x,0. For out these two critical stress states. completeness, we have included Figure 5 the extensive data set We present the data in Figures 3a and 3b to emphasize the on Berea sandstone of Khan et al. [ 1991 ], who picked the critical important connection between porosity change and AE activity. stress states by a combination of mechanical and AE We also plotted our data as AE rate versus axial strain and measurements. The choice of? and Q has the advantage that differential stress as is more typically done [Scholz, 1968]. In the these two variables are directly related to the first and second brittle regime the AE rate showed an accelerated increase near C' invariants of the stress tensor, respectively. However, it should and peaked in the postfailure region after the peak stress had be noted that the conventional triaxial test corresponds to a been attained (Figure 4a). Our observations are in qualitative particular loading path. For example, the Rothbach 1 experiment agreement with the more detailed AE studies on Berea sandstone shown in Figure 3a corresponds to a bilinear path in the P-Q by Locknet al. [1992] and on Darley Dale sandstone by Read space: the initial hydrostatic loading corresponds to a horizontal et al. [1995]. In the cataclastic flow regime, the AE rate showed path along the P axis up to 20 MPa and the subsequent an accelerated increase near C* and peaked after the sample had application of differential stress corresponds to a linear path with undergone a certain amount of shear-enhanced compaction a slope of 3:1. The values of œ and Q corresponding to C' (the (Figure 4b). For all the sandstones we investigated the AE rate open downward triangle in Figure 5) and to the peak stress (the was generally higher in the cataclastic flow regime, in agreement solid downward triangle) can be picked from the vertical axes in with the observation of Zhang et al. [1990a] on Berea sandstone Figures 3a and 4a, respectively. that compactive deformation is mor efficient in generating AE The peak stress data (solid symbols) map out the brittle failure tivity than its dilatant counterpart. envelope. The data for shear-induced dilation (open symbols) AE activity in rock is identified with damage processes, show that C' also has a positive correlation with mean stress, but including microcracking, frictional slip, and pore collapse. The the pressure sensitivity is somewhat less. Many previoustudies accurate location of hypocenter reveals the spatial complexity of have shown that the peak stress has a strong dependence on porosity, as well as secondary dependences on cementation and damage and provides important insights into the evolution of shear localization in brittle rocks. However, a number of technical challenges have to be overcome before the unambiguous determination of AE focal mechanism can be obtained [Locknet, 1993]. Consequently, the elucidation of the damage mechanisms requires the complementary input from microstructural observations. MenJndez et al. [1996] recently observed two distinct modes of damage evolution for shearinduced dilation and shear-enhanced compaction. Since very little intragranular cracking was observed before the peak stress had been attained, Mendndez et al. [1996] infer that in the prefailure stage (from C' to peak stress) the intense AE activity is primarily due to intergranular cracking (probably related to the shear rupture of lithified and cemented grain contacts) which allows the grains to move relative to one another, thus inducing an overall dilation of the pore space.!ntragranular cracks do not initiate until near the peak stress. Typically, they first develop in isolated clusters, and their subsequent coalescence results in shear localization in the postpeak stage. In contrast, intragranular cracking dominates the shear-enhanced compaction process throughouthe cataclastic flow regime. The surge in AE activity at the critical stress C* marks the onset of "Hertzian fractures" emanating from grain contacts. These intragranular cracks extend across the grains, resulting in comminution and pore collapse. Critical Stress States We have shown selected mechanical, AE, and microstructural data for the Adamswilier, Rothbach, and Berea sandstones respectively. Due to space limitation, we cannot detail here all the data accumulated for the six sandstones. Instead, we have identified and characterized several critical stress and

6 3014 WONG ET AL.' MECHANICAL DEFORMATION IN POROUS SANDSTONES Table 2. Compilation of Mechanical Data for Sandstone Samples Which Failed by Shear Localization Onset of Shear- Young's Pore Compaction Internal Dilatancy Hardening Factor s- Induced Dilation C' Peak Stress Modulus Compressibility Factor Friction Factor 0'3 -pp (MPa) ch- o' o' -cy., MPa E, GPa fl, 104 MPa 4 (MPa) A cp / Ae, Parameter fl hcr/e hmin/e - A P/AeP /t Adamswilier Berea 5! Boise II Darley Dale Rothbach (1) Rothbach (2) Kayenta I I clay content [Dunn et al., 1973; Logan, 1987; Vernik et al., compiled data are plotted in the P-Q stress space. For reference, 1993]. Presumably, these microstructural attributes may also the Mohr-Coulomb peak stresses are also included as solid exert similar influences on C'. However, our database is too symbols. For completeness, we included in Figure 6a the Berea limited for such correlation analysis. More extensivexperiments sandstone data of Jamison and TeufeI [1979], who reported an are desirable to address this important question. extensive data set on the transitional regime. Several important In the cataclastic flow regime we focus on the stress at the features should be noted. First, dilatant deformation is limited to onset of shear-enhanced compaction C*. Table 3 includes all our a relatively small subset of the stress space. This seems to be samples for which C* can be clearly determined from the characteristic of porous siliciclastic rocks. Second, the porosity and AE measurements. In Figures 6a and 6b the compactive yield envelopes for the onset of shear-enhanced compaction represents a negative correlation between P and Q at the critical states. Third, shear-enhanced compaction and BRITTLE FRACTURE cataclastic flow can occur over a broad range of stress conditions. The compactive yield envelope of Berea sandstone has stress values more than 5 times those of Boise sandstone. Since the 156 former has a porosity of 21% and the latter 35%, one may infer m that porosity exerts important influence over the compactire I yield behavior. However, other microstructural parameters may 199 [],,ll SANDSTONE C' peak V also be important since there is considerable discrepancy between Berea [] the yield envelopes of Bere and Kayenta sandstones, both of Darley Dale i,i A Rothbach 1 which have porosity of 21%. Last, data for Rothbach sandstone 59 Rothbach 2 cored in two orientations (parallel and perpendicular to bedding) Adamswilier suggest that the compactire yield stresses for Rothbach 2 are Boise I I somewhat higher. However, the mechanical data for this l] I.! sandstone (especially for Rothbach 1) show considerable variability from sample to sample. More systematic investigation EFFECTIUE MEAN STRESS P, MPa is desirable to confirm this trend. Figure 5. Peak stress (solid symbols) and C' (open symbols), the stress state at the onset of shear-inducedilation, are shown in Elastic Moduli and Compaction Rate the P (effective mean stress) and Q (differential stress) space. The data are compiled in Table 2. Note the positive slopes. We characterized the elastic and inelastic responses by two Additional Berea data of Khan et al. [1991] are also included. sets of parameters. In a triaxial compression test, the initial

7 WONG ET AL.' MECHANICAL DEFORMATION 1N POROUS SANDSTONES 3015 Table 3. Compilation of Mechanical Data for Sandstone Samples Which Failed by Compactive Cataclastic Flow Onsei of Shear-Enhanced Young's pore ComPaction Inelastic Compaction, C* Modulus Compressibility Factor Compaction Factor c - Pu, MPa. c 1 - ø.3, MPa E, GPa,8, 104.M. Pa -t AtI)/A, Aqvø/A p Adamswilier Berea Boise II Darley Dale Rothbach (1) O Rothbach (2) 1! ß Kayenta response to the application of differential stress is approximately constituents [Simmons and Wang, 1971]. We used the Voigtelastic. The Young's modulus E can be determined from the Reuss-Hill average [Hill, 1952] of an aggregate made up of initial slope of a plot of the differential stress versus axial strain quartz, feldspar, and calcite (with[3s = 2.65 x 104 Pa 4, 2.17 x (Figure 1). The independent determination of a second elastic 104 Pa 4, and 1.49 x 10 'n Pa 4, respectively) according to modulus is more complicated. If the pore pressure is maintained composition in Table 1. Over the pressure range of consideration constant, then the pore compressibility is defined to be here, the decrease of 13s with increasing pressure is expected to be [Zimmerman, 1991 ] of second order. The sum of the volume percentages of the three phases was normalized to 100%, and the contributions from minor minerals (clay, mica, and oxides) were neglected because of the paucity of elastic moduli data and uncertainty about their exact compositions. Using this procedure, we estimated the where VT i is the initial bulk volume of the sample (at room intrinsic compressibilities to be 2.60 x!0't Pa 4 (Adamswilier), conditions), V4 is the pore volume (under current effective 2.53 x 104 Pa 4 (Berea), 2.57 x 10 -n Pa 4 (Boise II), 2.54 x 10 ' pressure condition), and (I) is the current porosity value. The Pa 4 (Darley Dale), 2.56 x 104t Pa -x(kayenta) and 2.56 x 104 Pa 4 bracketed term corresponds to the slope of a plot of the porosity (Rothbach). The effective bulk moduli were then evaluated using change versus effective mean stress (Figures 2a and 2b) at the (2). initial application of differential stress (i.e., at the point where P Since the failure mode is sensitive to whether the inelastic - Pc-Pp). The elastic moduli E and [34 of selected samples deformation is dilatant or compactive, we also characterized the deformed in the brittle fracture and cataclastic flow regimes are porosity change behavior at the critical stresstates in terms of compiled in Tables 2 and 3, respectively. the "compaction factor" A /Ae, defined to be the ratio between Knowing the pore compressibility, the effective incremental change of the porosity (I) and axial strain. This compressibility (corresponding to the bulk response of the parameter was determined from the slope of curvesuch as those porous sandstone) its reciprocal (the bulk modulus K) can be in the bottom graph of Figure 1. In the brittle fracture and evaluated by [Walsh, 1965] cataclastic flow regime, ArI)/zX values were determined from the slopes at the peak stress and at the onset of shear-enhanced K-- / (13 + (V13,) (2) compaction, respectively. where it, denotes the intrinsic compressibility of the (porosity- The compaction factorso calculated represent the ratio of the free) solid matrix material which can be inferred from an total strains, which include the elastic and inelastic components. appropriate average of the elastic moduli of the mineral Since we have estimated the elastic modu!i, we can also estimate

8 3016 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES O ß II compactlue Berea yield enuelopes I Oarleg Dale I Kayenta I Rothbach 2 IRdamswiller O [] O O I Bolsell n Discussion Grain Crushing Pressure: A Quantitative Measure of Brittleness Determination of the critical stress and deformation parameters would not be possible without a fairly complete data set. In particular, the parallel measurements of porosity change and AE activity are crucial for this type of study. Several features of the failure mode and mechanical behavior should be noted. [] First, there are two distinct modes of failure: shear localization at t, oo a relatively low effective pressure or homogeneous cataclastic L o, '-... ' flow at elevated pressures. In the first mode, both the peak stress I 1 II 211 $ and C' show positive pressure dependence (Figure 5). As for the EFFECTIUE MEAN STRESS P, 1'4ra second mode, the inelastic deformation is associated with significant shear-enhanced compaction and work hardening, and (b) zig!o_ _e_._c.s_s co the differential stress for the onset of shear-enhanced compaction IPerpendlcuar(Z) [] ß decreases with increasing mean stress (Figure 6). Second, the parallel (1] O ß 15l sign of the compaction factor changes in the transition from brittle faulting to cataclastic flow. Typically, this transition C! occurs at an effective pressure Pbdt 0.15 P* (Figure 7). In the 190 transitional regime the behavior is more complex. A sample may have a positive value of A /Ae at the peak stress and yet fail by strain softening and development of conjugate shear bands. Third, since one end of the compactive yield envelope is "anchored" at a location of the P axis corresponding to œ*, this implies that if a rock requires a relatively high effective pressure I 51 I II for grain crushing to occur, then the critical stres states for the EFFECTIUE HEIIN STRESS P, MPa onset of shear-enhanced compaction are also expected to be high. Figure 6. (a) Stress state C* at the onset of shear-enhanced In other words, P* acts as a scaling parameter for the magnitudes compaction (open symbols) and peak stress for brittle fracture of the compactive yield stresses C* and the transition pressure (solid symbols) are shown in the P (effective mean stress) and Q Pbdt. This is illustrated by plotting the six yield envelopes from (differential stress) space. Note that the compactive yield Figure 6a in the normalized stress space, with coordinates envelopes have approximately elliptical shapes with negative P/P* and Q/P* (Figure 8). In this sense the critical pressure?* slopes. Data for six sandstones are shown. (b) Comparison of critical stresstates for Rothbach sandstone samples cored in two provides a quantitative measure of the "brittleness" of a porous siliciclastic rock. different directions. Samples perpendicular to bedding seem to have higher yield stresses. The data are compiled in Table an "inelastic compaction factor" by subtracting the elastic component from the total strain. At a given stres state we can determine the ratio between the differential stress and axial strain A(o]-o3)/A from the slope of curves such as those shown in the top graph of Figure 1. For a triaxial compression experiment with the conf'ming and pore pressures maintained constant, it can readily be shown that the inelasticompaction factor is given by z-1 dilatant ASSOCIATED FLOW RULE: -- parabolic envelope --- elliptic cap (Pc-Pf}/P* : gl A (A /A ) - 13,[A(.]-.3)/A ]/3 =. (3) A 1 - [6(0] -03)/6 ]/E Adamswilier ""' ', [] s.compactlue "-,, ',, Boise II Both the total and inelasticompaction factors are compiled Darley Dale in Tables 2 and 3, and the inelastic factors are plotted versus the Kayenta Rothbach 1 effective pressure (normalized to the critical effective pressure Rothbach 2 P*) in Figure 7. Experimentally determined compaction factors will be compared with plasticity theory predictions in a later section. It should be noted that 6 /6e > 0 and < 0 correspond to Figure 7. Inelasticompaction factor as function of effective compactire and dilatant processes, respectively. In the brittle pressure (normalized by the grain crashing pressure). Experimental dat are compared with theoretical predictions of faulting regime, since we determined the compaction factors at plasticity models assuming normality. The dashed curves are the peak stress for which A(o -o3)/ae = 0, the total and inelastic calculated from the associated flow rule for the two elliptical compaction factors are identical. The data for the sandstones (compactive yield) capshown in Figure 8. The solid curve is show very similar trends. Overall there is a monotonic increase from the associated flow rule for the parabolic envelope (for of AG/6 with increasing effective pressure. brittle fracture) shown in Figure 8.

9 ,,,, WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES 3017 parabolic envelope... elliptical cap I :1,' r.. I,=, o [1994] and the present study, we have also added four data sets on unconsolidated materials: Ottawa sand under elevated temperatures and pressures [Dewers and Hajash, 1995], coarse sand [Talwani et ai., 1973], and glass spheres [Garbrecht, 1973; Yin and Dvorkin, 1994]. Noting that the magnitude of P* is primarily controlled by the porosity and grain size, Zhang et al. [1990b] formulated a Hertzian fracture model that predicts ß (4) &,', with n -3/2. It can be seen from Figure 9 that our new compilation of data is in reasonable agreement with this t I theoretical prediction. In this micromechanical model the porous rock is idealized as a randomly packed assemblage of spherical I particles of several distinct sizes. Under hydrostatic loading, the NORHIILIZED EFFECTIUE HEIIN STRESS, neighboring grains are in normal contact and tensile stress Figure 8. Stress state C* at the onset of' shear-enhanced concentration is induced in the vicinity of the circle of contact compaction (open symbols) and peak stress for brittle fracture between two impinging grains [Johnson, 1985]. The maximum (solid symbols) are shown in the normalized P-Q space. tensile stress is attained at the perimeter of the contact area, and Effective mean stress and differential stress are both normalized the stress intensity factor K of a preexisting flaw located at the by the grain crushing pressure. Most of the C* datare bracketed contact region is approximated by that of an edge crack by the two elliptical caps (dashed curves), corresponding to (5) [Wilshaw, 1971]. Hertzian fracture can initiate if K reaches a with (¾,8) = (0.5, 0.5) and (0.5, 0.7), respectively. The peak stress critical value given by the fracture toughness Kt½. Assuming that datare fitted with a parabolic fracturenvelope (solid curve) the preexisting flaw dimension scales as the grain dimension R, corresponding to (7) with (Po, qo) = (0.49, 0.62). Key as in Figure 6. Zhang et al. [1990b] showed that this initiation condition would require the critical pressure P* to scale with the porosity and grain radius in accordance with (4). Since microstructural parameters (such as porosity and grain The Zhang et al. [1990b] model has been extended in two size) exert important control over the hydrostatic grain crushing recent studies. Without making any assumptions on the process [Zhang et al., 1990b], they should similarly influence the correlation between flaw and grain sizes, Brzesowslcy [1995] compactive yield envelope. We have updated our compilation of analyzed the influence of flaw statistics on grain crushing. Going P* as a function of initial porosity ß and grain radius R in Table beyond the initiation stage, Shah and Wong [1996] analyzed in 4. Other than new data for five sandstones from David et ai. some detail the propagation behavior of Hertzian fracture. Table 4. Compilation of Experimental Data on The Critical Effective Pressure for the Onset of Grain Crushing Under Hydrostatic Loading Material Reference Grain' ' Grain Crushing Radius Porosity Pressure R, mm ß œ*, MPa Glass sphere Glass bead Angular coarse sand Subrounded coarse Ottawa sand Ottawa sand (200øC) Ottawa sand Coarse sand Ottawa sand (150øC) Spheres Garbrecht [1973] Yin and Dvorkin [ 1994] Unconsolidated Sand Lee and Farhoomand [1967] Lee and Farhoomand [1967] Larnbe and Whitman [1969] Dewars and Hajash [1995] Zoback [1975] Talwani et al. [1973] Dewars and Hajash [1995] Consolidated Rocks this study et al. [1990b] et al. [1994] et al. [1994] et al. [1973] et al. [1990b] et al. [1990b] et al. [1994] Boise sandstone II Boise sandstone I Zhang Adamswilier David Rothb'ach sandstone David Lance sandstone Schock Kayenta sandstone Zhang St Peter sandstone Zhang Fontainebleau David Darley Dale sandstone this study Berea sandstone Zhang et al. [1990b] Yakuno basalt Shimada [1991] Oughtibridge ganister Hirth and Tullis [1989] > 1200

10 3018 WONG ET AL.: MECHANICAL DEFORMATION IN POROUSANDSTONES o ' i!! i Ills I _ i i"1 i till I I I! I lll I ' I I ONSET OF GRAIN CRUSHING: hydrostatic loading compaction, the grain contacts are subjected to oblique loading such that the resolved tangential component is somewhat less than the frictional resistance. The stress field and fracture mechanics of this scenario were analyzed recently by Shah and Wong [1996]. Their resultshow that the somewhat smaller enhancements in tensile stress and K are still adequate to initiate grain crushing at mean stresses significantly lower than P*, in agreement with the laboratory data shown in Figures 6 and I-I consolidated rock 'l unconsolidated sand ß glasspheres,,,,,,.i...,.,,i,,,... I porosity x grain radius R, mm Figure 9. Critical effective pressure for the onset of grain crushing under hydrostatic loading (/>*) as a function of the product of initial porosity ((I)) and grain radius (R). The data follow an approximately linear trend with a slope of-3/2 (equation (4)). The data are compiled in Table 4. Relation Between Compactive Yield Surface and Porosity Reduction The negative pressure dependence of the compactive yield envelope has been widely observed in soil mechanics [Chen, 1984; Desai and Siriwardane, 1984]. Therefore soil plasticity formulations such as the critical state [Schofield and Wroth, 1968] and cap [DiMaggio and Sandlet, 1971] models can provide a constitutive framework for the analysis of cataclastic flow. To use plasticity theory, we first need to identify an appropriate yield function. In critical state soil mechanics, bo circular and ellipticaloci in the P-Q space have been proposed.' For soil it is commonly assumed that the yield locus passes through the origin, but this restriction can be relaxed for cemented materials [Wood, 1990]. In the cap model an elliptical cap is commonly adopted [DiMaggio and Sandlet, 1971]. As shown in Figure 8, most of our normalizedata are bracketed by the ellipticaloci (dashed curves) given by (p/p,_¾)2 (Q/p,)2 + = 1 (1-¾) 2 82 Although additional complexities are incorporated into these with peaks at (% 5) = (0.5, 0.5) and (0.5, 0.7), respectively. Two models, the predictions of P* as a function of (I) and R are data points for Boise II and for Berea sandstones fall outside the qualitatively similar to (4) above. bounds of the two elliptical caps. The Boise samples failed at Our microstructural observations on Berea sandstone show relatively low stress values, and therefore measurements of C* that grain crushing and pore collapse are also the dominant may have significant errors. The two experiments for Berea were micromechanical processes in the cataclastic flow regime. The conducted at effective pressures very near P*. The differential initiation of grain crushing and pore collapse in a triaxially stress levels for compactive yield were relatively low, and the compressed sandstone are similar to those operative in values of the normalized data are sensitive to small variability of hydrostatically compacted samples, with microcracks radiated P* among samples. from grain contacts with a geometry reminiscent of tensile In plasticity theory the conventional approach is to associate indentation fractures. However, there is an important difference with the yield envelope (equation (5)) a flow rule which satisfies between the nonhydrostatically and the hydrostatically loaded the "normality" condition [Drucker, 1951]. We provide the samples. The stress-induced microcracking in a sample stressed mathematical details in an appendix. The normality condition beyond C* has a preferred orientation subparallel to c, whereas places restrictive constraints the plastic strain increments, and cracking in a sample hydrostatically compacted to beyond?* is specifically for axisymmetric loading it requires the inelastic relatively isotropic [Mendndez et al., 1996]. compaction factor to be given by the following expression: The anisotropy in microcracking indicates that the local stress field at a grain contact is strongly influenced by the AcI>P (6) nonhydrostatic loading. In addition to the normal loading AeP l+[8/(l_¾)12[(?_¾p,)/(3q)] induced by the mean stress, a tangential loading is induced by the The theoretical predictions for the compaction factor as a differential stresses. The tensile stress concentration at an function of the effective pressure Pc-Pp (normalized by the impingin grain contact may be significantly enhanced by such tangential loading. The end-member case with a tangential force crushing pressure P*) are shown as dashed curves in Figure 7. sufficiently high to overcome the frictional resistance on the They are associated with the two elliptical capshown in Figure contact surface (characterized by the friction coefficient f) was 8. On an elliptical cap the effective mean stress P is given by the analyzed by Hamilton and Goodman [1966], who showed that larger root of the equation AP'-- 2Bœ + C = 0, with the three the stress concentration the trailing end of the contact region is coefficients A= 9(1-7) enhanced by a factor of 15.5f This is probably an upper bound on the tensile stress enhancement, since relative grain movement C=9(1- ¾)2(Pc - Pp)2 + 2 (2¾- I)P '2, and was not evident in samples loaded to C*, implying that the stress by Q = 3 [P-(Pc- Pp)]. tangentialoading was not sufficiento cause slippage of the For hydrostatic loading the compaction factor attains its grain contact and therefore, at the onset of shear-enhanced 2 +82,B=9(1-T)2(Pc-Pp)+ 'TP*, the differential maximum value of 3 when the effective pressure equals the

11 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES 3019 critical crushing pressure P*. Although this the value to be expected of a relatively isotropic rock, we are not aware of any experiments in which the individual strain components were measured during hydrostatic compaction up to P*. At a lower AcI) p 3 effective pressure, a differential stress (Q > 0) is necessary for the -----= 3- (8) A P l+(2m/3)(p/ P*-Po) onset of shear-enhanced compaction, and the normality condition requires a compaction factor which is significantly lower than 3. where the effective mean stress P is given by the smaller root of The laboratory data are comparable in magnitude to the the equation AP 2 + BP + C = 0, with A = - /6/3 rn, B = /6/3(2mpo theoretical prediction for effective pressures down to 0.5 P* or so - 3), C = /6/3[(qo' mpo 2) ' 3(Pc-Pp)/t'*I, and the differential (Figure 7). Previous studies on carbonate rocks [Elliott and stress by Q = 3 [P- (Pc-Pp)]. The theoretical prediction for the Brown, 1985] and saline ice [Schulaon and Nickolayev, 1995] indicate similar agreement between experiment and theory. The compaction factor as a function of the effective pressure Pc- Pp (normalized by the crushing pressure P*) is shown as the solid minimum value of 0 for the inelastic compaction factor curve in Figure 7. It is associated with the parabolic failure (corresponding to the peak of the cap) is attained the effective envelope shown in Figure 8. pressure Pc - Pp = (¾-/5/ 3) P*. In the vicinity of this The normality condition requires the inelasticompaction effective pressure a sample fails in a transitional mode with an factor to be negative in the brittle regime, at effective pressures mount of inelasticompaction which is consistently greater than ranging from 0 to (Po- qo/3) P*. While the laboratory data do the theoretical prediction. The discrepancy suggests that the indicate dilatancy and therefore agree qualitatively with the associative flow rule and normality condition are not applicable theory, there is significant discrepancy between the magnitudes in the transitional regime. of the theoretically predicted and experimental values. This In summary, our data for the onset of shear-enhanced tendency for an associated flow rule to consistently overestimate compaction in six sandstones are in general agreement with the the magnitude of dilatation brittle failure has previously been plasticity models in two respects. First, the yield envelopes are noted in soil [Chen, 1984; Desai and Siriwardane, 1984] and in approximately elliptical, with stress magnitudes which scale as low-porosity rock [Senseny et al., 1983]. We could have used the grain crushing pressure P*. Second, the experimental some other mathematical equations to characterize the failure measurements of inelastic compaction are comparable to the envelope, but significant discrepancy between the associated predictions of the associative flow rule and normality condition dilatation and experimental observation is still expected unless a for effective pressures ranging from P* to P*. However, it nonassociative flow rule is adopted. should be noted that both the critical state and cap models also In their seminal paper on shear localization in pressureinclude very specific predictions on the strain hardening and sensitive, dilatant materials, Rudnicki and Rice [1975] compaction behavior which are not tested here. For Berea formulated an isotropic hardening model for rocks with three sandstone, Wong et al. [1992] showed that during cataclastic parameters characterizing the inelastic and failure behavior: an flow the yield envelope expands as a function of decreasing internal friction parameter g, a dilatancy factor 13, and a porosity, in qualitative agreement with the theoretical models. hardening modulus h. Mathematical details of the elastic-plastic Nevertheless, quantitative comparison with the theoretical constitutive equation and the scheme by which the three predictions would require detailed measurements of stress, strain parameters were extracted from experimental data are discussed and porosity changes during cyclic loading to "probe" the in the Appendix. The flow law is nonassociative provided that 13 expanding yield envelope. It is desirable to pursue a systematic Ix. It can be seen from the numerical values compiled in Table study of this question. 1 that 13 < Ix, implying the inelastic dilation was less than would be predicted by normality. Brittle Failure Envelope, Di!atancy, and Onset of Shear Rudnicki and Rice [1975] showed that the inception of Localization localization requires the hardening coefficiento attain a critical In the brittle fracture regime, the conventional approach is to value hcr given by characterize the peak stress with a linear failure envelope [Drucker and Prager, 1952]. However, experimental data over an extended range of effective pressures usually show a nonlinear hcr E -_ 18(1- (g- [3) v) ([3 + Ix + 3N) 2. (9) trend (Figure 5). A number of empirical equations [e.g., Hock and Brown, 1980; Vernik et al., 1993] have been proposed. As an For axisymmetric loading, N-- 1/ /3 and example, we consider the parabolic failure envelope used by implies that hcr is usually negative. Therefore shear localization Khan et al. [1991] to fit their Berea sandstone data. Taking into under triaxial compression can occur only if the sample account the scaling of the critical stresstates with P* (Figure 8), undergoestrain softening. The first term in the above expression we modify their equation by using the normalized stresses as the represents the influence of nonnormality, but it represents a variables: relatively small contribution to hcr, which is primarily controlled by the sum of 13 and g in the second term. Since the failure envelope is convex (Figure 5), the local slope and therefore tx p - Q qo = - m P -po. (7) decrease somewhat with increasing effective pressure. The overall dilatancy and therefore [3 also decrease with increasing The solid curve in Figure 8 corresponds to a parabolic envelope effective pressure. The net consequence is for her/e to become with the peak at (Po, qo ) = (0.49, 0.62). We have chosen m = qo / less negative with increasing effective pressure (Table 2). (l'po)2 so that the parabola when extrapolated to beyond the peak Acoustic emission [Locknet et al., 1992] and microstructural will intersect the horizontal axis at P = P*. Except for the Boise [Mendndez et al., 1996] observations on Berea sandstone indicate sandstone data, this failurenvelope provides a reasonable fit to that the onset of shear localization occurs in the post-failure stage, in qualitative agreement with the theoretical prediction. To the peak stress data in the brittle faulting regime. As elaborated in the appendix, the associated flow rule and normality condition imply that the inelasticompaction factor is given by the abovequation

12 ,,,, WONG ET AL.' MECHANICAL DEFORMATION IN POROUS SANDSTONES Table 5. Compilation of Experimental Data on The Critical Effective Pressure for the Transition From Brittle Faulting to Catac!astic Flow in Siliciclastic Rocks Porosity Grain Transition Rock Type Reference rp Radius Pressure R, mm Pbdt, MPa Heavitree quartzite Oughtibridge ganister Tennessee sandstone Tennessee sandstone Yakuno basalt Lance sandstone Gosford sandstone Hirth and Tullis [1989] Hadizadeh and Rutter [1983] Rutter and Hadizadeh [ 1991 ] Scott and Nielson [199I] Shimada [ 1991 ] Schock et al. [1973] Adamswiller sandstone this study Bunter sandstone Darley Dale sandstone this study Berea sandstone Berea sandstone Edmond and Paterson [1972] Gowd and Rumreel [1980] this study Jamison and Teufel [! 979] Kayenta sandstone Wong et ai. [1992] Rothbach sandstone Boise II sandstone this study this study > > I ! I pinpoint the onset of localization would require very detailed microstructural observations and acoustic emission locations of all our samples. Since this is well beyond the scope of the present study, we will instead try to place a lower bound on hcr by determining the maximum negative slope in the post-failure stage in each sample which failed by the development of a thoroughgoing shear band. The corresponding hardening coefficient is denoted by h,,in, and the numerical data are compiled in Table 2. The laboratory data show thathe slope of the stress-strain curves in the postfailure stage is more gentle at a {a} : I I ll[,% -%,... I... I lb} i i i, I,,, I! 1888 _ 199 -x ''., '"'"-, cataclastic flou brittle faulting x '"" xx [] x xx,v X,, E] tectonic "', ', faulting. le IO x )m: < x 'x, ',,,,,,, 1 failure modes of siliciclastic rocks [] cataclastic flow X brittle faulting 8. I... '... '-( :-'... ' ! 8.1 porosity x grain radius x ß R, mm.=- i 8.1 Enu'ada & Navajo ss, I::[:! Wingate sandstone, San Rafael desert, UT Colorado Punchbowl fault North Scapa sandstone sandstone, CA Orkney, Scotland Moine Cambrian thrust quartzite, zone i Lyons Boulder sandstone, fault, CO porosity x grain radius R, mm Figure 10. (a) The failure modes at fixed effective pressures of 14 siliciclastic rocks are plotted versus the product of initial porosity ( ) and grain radius (R). The experimental datare compiled in Table 5. Effective pressures for the. transition from brittle fracture to cataclastic flow are bracketed by the tw dotted lines with slopes of-3/2, corresponding to (10).(b) Effective overburden pressures for occurrence of tectonic faulting siliciclastic rock formations are plotted versus the product of porosity (tp) and grain radius (R). The dat are compiled in Table 6. The pressure is estimated from the overburden depth assuming hydrostatic pore pressure in a compressional setting. For reference the dotted lines for the transition pressures from Figure 10a are included. The geologic datall fall in the brittle faulting regime, in agreement with the laboratory data.

13 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES 3021 higher effective pressure (Figure 1), and therefore magnitude of h decreases with increasing effective pressure. This trend is in qualitative agreement with the theoretical prediction her as a function of the effective pressure (Table 2). However, there is significant difference between the theoretically predicted her and experimentally observed lower bound hmi,,, with the former being less by as much as 1 order of magnitude. siliciclastic rocks with porosities ranging from <1% to 35%. Although many of the classical studies on the low-temperature brittle-ductile transition were conducted over limited ranges of effective pressure and did not measure the porosity changes or AE activities, they usually include qualitative characterization of the failure mode as a function of effective pressure. Rutter and Hadizadeh [1991] provided a relatively comprehensive The discrepancy suggests that the model has not captured all compilation, from which we obtained most of the data for Table the important features of constitutive behavior responsible for the 5. We have added data from the present study and plotted the inception of shear localization in the brittle regime. Stress- transition pressure versus R in Figure 10a. Two dashed lines induced anisotropy in the elastic and inelastic deformation is (with slope of-3/2 on the log-log plot) bracket the effective neglected. The isotropic hardening model assumes thathe initial pressures over which the failure mode transition occurs, in and subsequent yield surfaces are smooth, but micromechanical reasonable agreement with the prediction of equation (10). considerations suggest that vertex-like structures (corners) may The empirical relation (10) can also be tested with geologic develop in subsequent yield surfaces. These processes are data. We have compile data on tectonic faulting in six sandstone alestabilizing and tend to decrease the magnitude of her [Rudnicki formations in Table 6. These studies are chosen because they all and Rice, 1975; Rudnicki, 1977; Chau and Rudnicki, 1990]. A include characterization of porosity, grain size, and overburden constitutive equation incorporating these complexities may depth z. Different styles of faulting were involved. In some cases provide predictions the onset of shear localization which are the initial faulting style is poorly constrained. To establish in better agreement with experimental observations. However, contact with the laboratory data, we estimate the effective characterization of the additional constitutive parmeters overburden pressure by (p-pw)gz, with saturatedensity p -- necessitate the use of loading configurationsuch as plane strain 2250 kg m '3 and water density Pw = 1000 kg m '. This choice of [e.g., Ordet al., 1991] and rotary shear [e.g., Olsson, 1992], densities is appropriate for a relatively porous sandstone at which are not axisymmetric. hydrostatic pore pressure. Effective overburden pressure would be lower if the pore pressure is super-hydrostatic. In a Critical Pressure for the Transition From Brittle Fracture compressional tectonic environment, the effective overburden to Compactive Cataclastic Flow pressure is analogous to the effective pressure ch - Pp in a triaxial compression experiment. For strike-slip and extensional settings, Our experimental data indicate that the transition from brittle faulting to compactive cataclastic flow occurs at a critical it corresponds to an upper bound on ch - Pp. Effective overburden pressures are plotted versus R in effective pressure (Pbdt) that scales as the grain crushing pressure Figure 10b. If the experimentally based criterion for brittle- P*, which is primarily controlled by the initial porosity and grain ductile transition is applicable to tectonic faulting, then the size (equation(4)). Hence one would expect the porosity and effective overburden pressures for the occurrence of brittle grain size to exert similar influence over the transition pressure faulting (inferred from geologic data) should fall below the Pbdt : transition boundary (indicated by the dashed lines in Figure 10b). at ( R ) ( o ) Indeed, the geologic data all fall within the brittle faulting regime as inferred from the laboratory data. In this sense, the tectonic with n -3/2. This empirical relation is tested with data on 13 faulting and experimental data are in agreement, implying that Table 6. Compilation of Geologic Data on the Overburden Depth of Tectonic Faulting in Siliciclastic Rock Formations as a Function of Porosity and Grain Size Fault Zone/Rock Type Reference Grain Overburden Radius Porosity Depth, R (mm) ß km Colorado National Monument Wingate sandstone San Rafael Desert, Utah Entrada sandstone Navajo sandstone Moine thrust zone, Scotland Cambrian quartzite Jarnison and Stearns [1982] Aydin [ 1978], Stokes [ 1986] Aydin [1978], Stokes [1986] Punchbowl fault zone, Calilbmia Sandstone Chester and Logan [1986] Boulder fault, Colorado Lyons sandstone,4nders and?iltschko[1994] North Scapa fault, Orkney, Scotland North Scapa sandstone Hippler [ 1993] > Blenkinsop and Rutter [1986] * ~ 0.!25 # 0.05 < , I < 2.5 * Estimated from micrograph of the fault rock in the publication. # Inferred from the authors' description of the sandstone as "fine-to-medium-grained."

14 3022 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES (10) provides a first-order description of the influence of porosity 4. In the brittle faulting and transitional regimes, the and grain size on the transition from brittle faulting to cataclastic associated flow rule predicts dilatancy to increase with flow in geologic settings. A more thorough comparison would require geologic data over a broaderange of overburden depths. This study focuses on the control of porosity and grain size on the low-temperature brittle-ductile transition in laboratory and tectonic environments. However, we are not suggesting that other decreasing effective pressure. Though in qualitative agreement with experimental observations on porosity change, the theoretical predictions are significantly higher in magnitude. Rudnicla' and Rice's [!975] localization analysis predicts the onset of shear localization at a Critical hardening coefficiem microstructural attributes (such as cementation, clay content, which is significantly more negative than experimental data sorting, and preexisting anisotropy) do not influence the inelasticity and failure mode. It should also be noted that the secondary influences of temperature and strain rate on the brittle Better agreement may be provided by a constitutive model which accounts for stress-induced anisotropy and vertex-like features on the subsequent yield envelopes. failure processes have been neglected. At this point there is a paucity of data pertaining to these questions, and further Appendix: Plasticity Theory and Localization systematic study would be helpful toward the physical Analysis understanding of these effects in a typical geologic setting. Our results are also applicable to geotechnical applications. In reservoir engineering, the stability of a deep borehole has been found to be sensitively dependent on the extent of work hardening induced by shear-enhanced compaction [e.g., For an isotropic material the initial yield behavior can be characterized in the stress (cvu) space as a function of three stress invariants [Fung,!965]. In particular, a yield envelope (empirically determined by conventional triaxial tests) which is Santorelli et al., 1986; Veeken et al., 1989]. Perforation damage dependent only on the effedtive mean stress? and differential. commonly occurs by grain crushing [Papamichos et al., 1993]. stress Q can be generalized to a yield function dependent on the In percussive and rotary drilling, the compaction zone modifies first and second stress invariants the stress field in the proximity of the indenter and reduces the efficiency of the overall drilling process [Miller and Cheatham, 1972; Suarez-Rivera et al, 1990]. The rock deformation data summarized here provide important constraints on the formulation of constitutive models for other geological problems. Continuum plasticity models have J2 = [(c 11-c 22 )2 +(cv22-1x33 )2 +(C 33--CVll )2]/6 +c '232 +c 132 (A1) been adopted for the analyses of carbonate [Brown and Yu, 1988], shale [$teiger and Leung, 1991], sediment [Jones and which are related to P and Q by I]=3P and 3J2=Q 2. Addis, 1986] and fault gouge [Scott et al., 1994] deformation. Consequently an elliptical cap given by (5) for axisymmetric Rutter and Neumann [1995] suggested that the extraction of granitic magma involves a deformation mechanism analogous to loading corresponds to the following yield function in the general stress space: shear-enhanced compaction. Our results on the applicability and limitation of associative and nonassociative plasticity models to sandstone may provide useful insights on deformation processes f(cv/j) = (II-C)2.2 + = 0 ½2) in these other geomaterials. Conclusion with a = 3 (1-¾) P*, b = 8/ /3 P*, and c = 3¾ P*. At the onset of shear-enhanced compaction, if the inelastic deformation follows Drucker's [1951] postulate of material stability, then e 1. The transition from brittle faulting to compactive cataclastic following "normality" condition applies: flow in a porous sandstone occurs at an effective pressure that scales as the grain crushing pressure. The transition pressure decreases with increasing porosity and grain size in accordance a.f. Az = ac /d with a power law. This implies that a more porous and coarsegrained rock tends to be less brittle, and the grain crushing where A denotes the plastic increment of the strain tensor and pressure represents a quantitative measure of brittleness. A is a positive scalar. Substituting the yield function (A2) in o Geologic data on tectonic faulting in si!iciclastic formations (of (A3), we obtain the ratio between the volumetric and xi l different porosity and grain size) are consistent with the strains: laboratory data. 2. In the brittle faulting regime, shear-induced dilation Aei = 1862(I1 -c) - initiates in the prepeak stage at a stress level C'. Both C' and the A$1P 1 a2(2611 -cv22 -c 33)+662(I1 -c) (A4) peak stress increase with effective mean stress. In the cataclastic flow regime, inelastic deformation is associated with strain The plastic strain components are related to the inelastic strains hardening and shear-enhanced compaction, initiating at a stress estimated from our laboratory data. Since the plasticomponent level C* which decreases with increasing effective mean stress. of volumetric strain is dominated by porosity change, we have Critical stress levels C' and C* are both marked by surges in AE A e A '. The axial component of plastic strain is simply activity. 3. The compactive yield stress C* maps out an approximately Aesp 1 =Ae '. Substituting into (A4), the inelastic elliptical yield envelope, in agreement with the critical state and factor can be expressed in the specific form for triaxial cap models. Magnitude of C* increases with decreasing porosity compression given by (6) in the text. and grain size. Porosity changes predicted by the flow rule In the brittle fracture regime the parabolic failurenvelope associated with the yield envelope are in reasonable agreement given by equation (8) for axisymmetric loading can with experimental data in the cataclastic flow regime. generalized to a yield function in terms of the first and compaction

15 , WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES 3023 stress invariants: f( /j) = CZo-0:1-0:2 = 0 (^5) where Cto +3ctl +9(z2 =0' with 0:o=Xfg/3(qo-mpo), tests el=2 /'g/9mpo and 0:2 =-xfg)27m' Athe peak stress, if the inelastic deformation follows Drucker's [1951] postulate of material stability, then the normality condition (A3) implies that the ratio between the (plastic) volumetric and axial strains is given by zse _ 3-6 (A6) AglP 1 2- x/- (0:1 + 20:211) If we assume that Ae/ =AO p, then the inelastic compaction factor can be expressed in the specific form for triaxial compression given by (8) in the text. In the brittle regime, a nonassociative elastic-plastic constitutive equation was proposed by Rudnicki and Rice [ 1975]. Initiation of inelastic deformation is governed by the yield criterion f (c ij ) = xf ll - k =0 (^7) sandstone the slope was determined by consideration of other data for peak stress from Wong et al. [1992] not included in Table 2. It should be noted that g as defined by Rudnicki and Rice [1975] is less than tan {, the "coefficient of internal friction" commonly associated with the Mohr-Coulomb failure envelope. As a matter of fact, it can be shown that for triaxial compression 2x/' ' sin q> =. (A O) 3 - sin q0 After some algebraic manipulation, it can also be shown that the hardening parameter is given by where the tangent modulus h,an corresponds to the local slope of a plot of the differential stress versus axial strain (such as Figure!) in a conventional triaxial test. In this study we estimated g and [3 at the peak stress for all samples which failed by shear localization, including those in the transitional regime which failed by development of conjugate shear bands showing strain softening (h < 0) and slightly positive compaction ([3 < 0). Acknowledgments. We are grateful to Joanne Fredrich, Daniel Jeannette, and Phil Meredith for providing the Boise, Vosges, and Darley Dale sandstone samples, respectively. Experiments on the were g and k are parameters characterizing the internal friction Kayenta sandstone were performed by Jiaxiang Zhang, and some of the and cohesion, respectively. This yield function is identical to that modal and grain size analyses were conducted by Ann Cox and Daniel proposed by Drucker and Prager [1952], and if the normality Jeannette. We have benefited from discussions with Yves Bemab6, condition is imposed, then the inelastic compaction factor is Joanne Fredrich, Beatriz Men6ndez, Gene Scott, Ketan Shah, and Larry given by -3g/( /3- g). As discussed in the text, the associated Teufel during variou stages of this project. We thank Mike Batzle, John Rudnicki, and Steve Mackwell for their critical reviews. The second flow rule overestimates the rock's potential to dilate. To better author was supported by Elf Aquitaine and the French Minist re des characterize porosity change, a dilatancy fhctor 13 (corresponding Affaires Etrang res on a postdoctoral fellowship during his stay at Stony to the ratio of the plastic volumetric strain to the shear strain) is Brook. This research was supported by the Office of Basic Energy incorporated the nonassociative formulation. The strain Sciences, Department of Energy under grant DEFG0294ER14455 and by NATO under grant CRG hardening (and softening) behavior is characterized by a parameter h, whose sign is positive (or negative) for the prepeak (or postpeak) stage. As elaborated by Rudnicki [1984], the References parameters g, 13, and h evolve with the inelastic deformation and Anders, M.H., and D.V. Wiltschko, Microfracturing, paleostress, and the the plastic strain tensor is described by growth of faults, J. Struct. Geol., 16, , Antonellini, M., and A. Aydin, Effect of faulting on fluid flow in porous I G ' 1(' t :i5 =1 ij {Jkl }.t A(C kk +pp (A8) h (2 2 + ij 2Xf 2 AIj kl +' with the deviatoric stress tensor c 0' = cv./i - (o / 3) 8/j. In the abovequation we have followed Rudnicki and Rice's [1975] convention, taking tensile stresses to be positive. The constitutive parameters can be extracted from triaxial compression experiment data in the following manner. If we assume that zx A O p, then the dilatancy factor is given by [3 = - /x p/ Ag p (3-A p / Ag p) )1 Antonellini, sandstones: M., Petrophysical A. Aydin, properties, D.D. AAPG Pollard, Bu!l., 78, Microstructure , of deformation bands in porous at Arches National Park, (A9) The difference in sign convention between the dilatancy and compaction factorshould be noted: if AO p / AaP < 0, then [3 > 0. According to (A7), the friction parameter g can be evaluated as /3/3 times the slope of the failure end, elope in the P-Q space (Figure 5). For Berea sandstone, we estimated the local slope of a parabola failure envelope (equation (7)) fitted to the laboratory data. For the other samples with relatively few data points, we simply fitted a straight line to obtain a constant g. For Kayenta Utah, J. Struct. Geol., 16, , Aydin, A., Small faults formed as deformation bands in sandstones, Pure Appl. Geophys., 116, , Aydin, A., and A.M. Johnson, Development of faults as zones of deformation bands and as slip surfaces in sandstone, Pure Appl. Geophys, 116, , Bemab6, Y., and W.F. Brace, Deformation and fracture of Berea sandstone, in The Brittle-Ductile Transition in Rocks, Geophys. Monogr. Serf. Vp;/56, edited by A.G. Duba et al., pp , AGU, Washington, D.C., Blenkinsop, T.G., and E.H. RutteL Cataclastic deformation of quartzite in the Moine Thrust Zone,,L. Struct. Geol., 8, , Brace, W.F., Volume changes during fracture and frictional sliding: A review, Pure Appl. Geophys., 116, , Brown, E.T., and H.S. Yu, A model for the ductile yield of porous rocks, lnt. J. Numer. Anal. Methods Geomech., 12, , Brzesowsky, R., Micromechanics of Sand Grain Failure and Sand Compaction, Ph.D. thesis, Utrecht Univ., Utrecht, Netherlands, Chau, K.-T., and J.W. Rudnicki, Bifurcations of compressible pressuresensitive materials in plane strain tension and compression, or. Mech. Phys. Solids, 38, , Chen, W.F., Soil mechanics, plasticity and landslides, in Mechanics of Inelastic Materials, edited by G.J. Dvorak, and R.T. Shield, pp , Elsevier, Amsterdam, 1984.

16 3024 WONG ET AL.: MECHANICAL DEFORMATION IN POROUS SANDSTONES confining pressure, I, Yield and failure surfaces, and nonlinear elastic Chester, F.M., and J.M. Logan, Implications for mechanical properties of brittle faults from observations of the Punchbowl fault zone, response, Int. J. Plasticity, 7, , Califomia, P./IGEOPH, 124, , Lambe, T.W., and R.V. Whitman, Soil Mechanics, John Wiley, New Curran, J.H., and M.M. Carroll, Shear stress enhancement of void York, compaction, o r. Geophys. Res., 84, , Lee, K.L., and I. Farhoomand, Compressibility and crushing of granular David, C., T.-f. Wong, W. Zhu, and J. Zhang, Laboratory measurement soil in anisotropic triaxial compression, Can. Geotech. J., 4, 68-99, of compaction-induced permeability change in porous rock: Implications for the generation and maintenance of pore pressure Lockner, D., The role of acoustic emission in the study of rock fracture, excess in the crust, Pure Appl. Geophys., 143, , Int. J. Rock Mech. Min. Sci., 30, , Desai, C.S., and H.J. Siriwardane, Constitutive Laws for Engineering Locknet, D.A., J.D. Byeflee, V. Kuksenko, A. Ponomarev, and A. Materials, 468 pp., Prentice-Hall, Englewood Cliffs, N.J., Sidorin, Observations of quasistatic fault growth from acoustic Dewars, T., and A. Hajash, Rate laws for water-assisted compaction and emissions, in Fault Mechanics and Transport Properties of Rocks, stress-induced water-rock interaction sandstones, o r. Geophys. Res., edited by B. Evans and T.-f. Wong, pp. 3-32, Academic, San Diego, 100, , Calif., DiMaggio, F.L., and I.S. Sandlet, Material model for granular soils, o r. Logan, J.M., Porosity and the brittle-ductile transition Eng. Mech. Div., Am. Soc. Civ. Eng., 97, , rocks, in Physics and Chemistry of Porous Media II, AlP Conf. Proc., Drucker, D.C., A more fundamental approach to plastic stress strain 154, , relations, in Proceedings of the First U.S. National Congress Applied Men6ndez, B., W. Zhu, and T.-f. Wong, Micromechanics of brittle Mechanics, pp , faulting and cataclastic flow in Berea sandstone, o r. Struct. Geol., 18, Drucker, D.C., and W. Prager, Soil mechanics and plastic analysis of 1-16, limit design, Q. Appl. Math., 10, , Miller, T.W., and J.B. Cheatham, Analysis of the indentation of a Dunn, D.E., L.J. LaFountain, and R.E. Jackson, Porosity dependence and mechanism of brittle fracture in sandstones, J. Geophys. Res., compacting material by a perfectly rough wedge, Int. J. Rock Mech. Min. Sci., 9, , , , Olsson, W.A., The formation of a yield-surface vertex in rock, 'm Edmond, J.M., and M.S. Paterson, Volume change during the Proceedings of the 33rd U.S. Symposium on Rock Mechtm., deformation of rocks at high pressure, Int. J. Rock Mech. Min. Sci., 9, edited by J.R. Tillerson, and W.R. Wawersik, pp , A. A , Elliott, G.M., and E.T. Brown, Yield of a soft, high porosity rock, Balkema, Rotterdam, Netherlands, Ord, A., I. Vardoulakis, and R. Kajewski, Shear band formation in. Geotechnique, 35, , Gosford sandstone, Int. J. Rock Mech. Min. Sci., 28, , Fischer, G.J., and M.S. Paterson, Dilatancy during rock deformation high temperatures and pressures,,/. Geophys. Res., 94, , Fredrich, J.T., B. Evans, and T.-f. Wong, Micromechanics of the brittle to plastic transition in Cartara marble, o r. Geophys. Res., 94, , Ftmg, Y.C., Foundations of Solid Mechanics, 525 pp., Prentice-Hall, Papamichos, E., I. Vardoulakis, and H. Ouadfel, Permeability reduction due to grain crushing around a perforation, Int.,1. Rock Mech. Min. Sci., 30, , Paterson, M.S., Experimental Rock Deformation-The Brittle Field, 254 pp., Spinget-Verlag, New York, Read, M.D., M.R. Ayling, P.G. Meredith, and S.A.F. Murrell, Microcracking during triaxial deformation of porous rocks monitored Englewood Cliffs, N.J., by changes in rock physical properties. II. Pore volurnometry and Garbrecht, D., Kornbruch in irregularen Haufwerken aus elastisch- acoustic emission measurements on water-saturated rocks, spr6den KugeIn, Heft 56, Inst. fur Bodenmechanik und Tectonophysics, 245, , Felsmechanik, Univ. Fridericiana, Karlsruhe, Germany, Gowd, T.N., and F. Rummel, Effect of confining pressure on fracture behavior of a porous rock, Int.,I. Rock Mech. Min. Sci., 17, , Rudnicki, J.W., The effect of stress-induced anisotropy on a model of brittle rock failure as localization of deformation, in Energy Resources and Excavation Technology, Proc. 18th U.S. Symposium on Rock Mechanics, pp. 1-8, Keystone, Colo., Hadizadeh, J., and E.H. Rutter, The low temperature brittle-ductile Rudnicki, J.W., A class of elastic-plastic constitutive laws for brittle transition in a quartzite and the occurrence of cataclastic flow in nature, Geol. Rundsch., 72, , Hamilton, G.M., and L.E. Goodman, The stress field created by a circular sliding contact, o r. Appl. Mech., 33, , rocks, o r. Rheology, 28, , Rudnicki, J.W., and J.R. Rice, Conditions for the localization of deformation in pressure sensitive dilatant materials, o r. Mech. Phys. Solids, 23, , Handin, J., R.V. Hager, M. Friedman, and J.N. Feather, Experimental Rutter, E.H., and J. Hadizadeh, On the influence of porosity on the lowdeformation of sedimentany rock under confining pressure: Pore pressureffects, AAPG Bull., 47, , temperature brittle-ductile transition in siliciclastic rocks,/. Struct. Geol., 13, , Hill, R., The elastic behavior of a crystalline aggregate, Froc. Fhys. Soc., Rutter, E.H., and D.H.K. Neumann, Experimental deformation of.465, , partially molten Westerly granite under fluid-absent conditions, with Hippier, S.J., Deformation microstinctures and aliagenesis in sandstone implications for the extraction of granitic magmas, J. Geophys. Res., adjacento an extensional fault: Implications for the flow and 100, , entrapment of hydrocarbons, IAPG Bull., 77, , Santorelli, F.J., E.T. Brown, and V. Maury, Analysis of borehole stresses Hirth, G., and J. Tullis, The effects of pressure and porosity on the using pressure-dependent, linear elasticity, Int. J. Rock Mech. Min. micromechanics of the brittle-ductile transition, o r. Geophys. Res., 94, Sci., 23, , , Schock, R.N., H.C. Heard, and D.R. Stevens, Stress-strain behavior of a Hirth, G., and J. Tullis, The brittle-plastic transition in experimentally granodiorite and two graywackes on compression to 20 kilobars, d. deformed quartz aggregates, o r. Geophys. Res., 99, , Geophys. Res., 78, , Schofield, A.N., and C.P. Wroth, Critical State Soil Mechanics, 310 pp., Hock, E., and E.T. Brown, Underground Excavation Engineering, Inst. McGraw Hill, New York, of Mining and Metal., London, Scholz, C.H., Microfracturing and the inelastic deformation of rock in Jamison, W.R., and D.W. Steams, Tectonic deformation of Wingate compression, o r. Geophys. Res., 73, , Sandstone, Colorado National Monument, A tpg Bull., 66, Schulson, E.M., and O.Y. Nickolayev, Failure of columnar saline ice 2608, under biaxial compression: Failurenvelopes and the brittle4uctile Jamison, W.R., and L.W. Teufel, Pore volume changes associated with transition, o r. Geophys. Res., 100, , failure and frictional sliding of a porous sandstone, Froc. U.S. Rock Scott, D.R., C.J. Marone, and C.G. Sammis, The apparent friction.of MechSymp.., 20,!63-170, granular fault gouge in sheared layers, o r. Geophys. Res., 99, 723!- Johnson, K.L., Contact Mechanics, 452 pp., Cambridge Univ. Press, 7246, New York,!985. Scott, T.E., and K.C. Nielson, The effects of porosity on the bd e- Jones, M.E., and M.A. Addis, The application of stress paths and critical ductile transition in sandstones, o r. Geophys. Res., 96, , state analysis to sediment deformation, o r. Struct. Geol., 8, , Senseny, P.E., A.F. Fossum, and T.W. Pfeifle, constitutive laws for low-porosity rocks, Int.,i. Numer. n Khan, A.S,, Y. Xiang, and S. Huang, Behavior of Berea sandstone under Methods. Geornech., 7, , in sedimentary Non-associate

17 WONG ET AL.' MECHANICAL DEFORMATION IN POROUS SANDSTONES 3025 Shah, K.R., and T.-f. Wong, Fracturing at contact surfaces subjected to normal and tangential loads, Int. J. Rock Mech. Min. Sci., in press, Wong, T.-f., Mechanical compaction and the brittle-ductile transition in porous sandstones, in Deformation Mechanisms, Rheology and Tectonics, edited by R.J. Knipe and R.H. Rutter, pp , Geol. Society Special Publication Vol. 54, London, Shimaria, M., Comment on "Theffects of pressure and porosity on the micromechanics of the brittle-ductile transition in quartzite" by Greg Wong, T.-f., H. Szeto, and J. Zhang, Effect of loading path and porosity Hirth and Jan Tullis, o r. Geophys. Res., 96, , on the failure mode of porous rocks, Appl. Mech. Rev., 45, , simmons, G., and H.F. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties, MIT Press, Cambridge, Mass., Wood, D.M., Soil Behaviour and CritiCal State Soil Mechanics, 462 pp., 197I. Cambridge Univ. Press, New York, Steiger, R.P., and P.K. Leung, Critical state shale mechanics, in Rock Yin, H., and J. Dvorkin, Strength of cemented grains, Geophys. Res. Mechanics as Multidisciplinary Science, edited by J. Roegiers, pp. Lett., 21, , , A.A. Balkema, Rotterdam, Netherlands, Zhang, J., T.-f. Wong, and D.M. Davis, High pressurembrittlementt Stokes, W.L., Geology of Utah, Utah Mus. of Nat. Hist. and Utah Geol. and shear-enhanced compaction in Berea sandstone: Acoustic and Miner. Sur., Salt Lake City, emission measurement and microstructural observation, in Rock Suarez-Rivera, F.R., N.G.W. Cook, G.A. Cooper, and Z. Zheng, Mechanics Contributions and Challenges, Proceedings of 31st U.S. Indentation by pore collapse in porous rocks, in Rock Mechanics Symposium on Rock Mechanics, edited by W.A. Hustmlid, and G.A. Contributions and Challenges, Proceedings of the 31st U.S. Johnson, pp , A.A. Balkema, Rotterdam, Netherlands, 1990a. Symposium on Rock Mechanics, edited by W.A. Hustrulid and G.A. Johnson, pp , Balkema, Rotterdam, Netherlands, Zhang, J., T.-f. Wong, and D.M. Davis, Micromechanics of pressure- Talwani, P., A. Nur, and R.L. Kovach, Compressional and shear wave induced grain crashing in porous rocks, J. Geophys. Res., 95, 341- velocities in granular materials to 2.5 kilobars, J. Geophys. Res., 78, 352, 1990b , Zhang, J., T.-f. Wong, T. Yanagidani, and D.M. Davis, Pressure-induced Teufel, L.W., D.W. Rhett, and H.E. Farrell, Effect of reservoir depletion microcracking and grain crashing in Berea and Boise sandstones: and pore pressure drawdown on in situ stress an deformation in the Acoustic emission and quantitative microscopy measurements, Elcorisk field, North Sea, Proc. U.S. Rock Mech. Symp., 32, 63-72, Mech. Mater., 9, 1-15, 1990c Zimmerman, R.W., Compressibility of Sandstones, 173 pp., Elsevier, Tullis, J., and R.A. Yund, The brittle-ductile transition in feldspar New York, aggregates: An experimental study, in Fault Mechanics and Zhu, W., and T.-f. Wong, The transition from brittle faulting to Transport Properties of Rocks, edited by B. Evans and T.-f. Wong, cataclastic flow: Permeability evolution, or. Geophys. Res., this issue. pp , Academic, San Diego, Calif., Zoback, M.D., High pressure deformation and fluid flow in sandstone, Underwood, E.E., Quantitative StereoIogy, 274 pp., Addison-Wesley, granite and granular materials, Ph.D. thesis, Stanford Univ., Stanford, Reading, Mass., Calif., Veeken, C.A.M., J.V. Walters, C.J. Kenter, and D.R. Davis, Use of plasticity models for predicting borehole stability, in Rocks at Great Depth, vol. 2, edited by V. Maury and D. Fourmaintraux, pp 'l avid, institut de Physique du Globe, Strasbourg, France ( 844, A.A. Balkema, Rotterham, Netherlands, christian.david@eopgs-ulp.u-strasbg.fr) Vemik, L., M. Bruno, and C. Boyberg, Empirical relations between T.-f. Wong and W. Zhu, Department of Earth and Space Sciences, State University of New York at Stony Brook, Stony Brook, NY compressire strength and porosity of siliciclastic rocks, Int. 1. Rock Mech. Min. $ci., 30, , ( wong seisml.ess.sunysb.edu; zhu seisml.ess.sunysb.edu) Walsh, J.B., The effect of cracks on the compressibility of rocks, J. Geophys. Res., 70, , Wilshaw, T.R., The Hertzian fracture test, J. Phys., D4, , Received January 5, 1996; revised August 5, 1996; accepted October 21, 1996)