shear-enhanced compaction and permeability...
TRANSCRIPT
ELSEVIER Mechanics of Materials 25 (1997) 199-214
MECHANICS OF
MATERIALS
Shear-enhanced compaction and permeability reduction: Triaxial extension tests on porous sandstone
Wenlu Zhu a,*, Laurent G.J. Montesi b, Teng-fong Wong a a Department of Earth and Space Sciences, State Unioersity of New York, Stony Brook, NY 11794-2100, USA
b Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 21 June 1996; received in revised form 7 February 1997; accepted 11 February 1997
Abstract
Triaxial extension experiments were conducted to investigate the influence of radial stress on porosity and permeability (for hydraulic flow along the axial direction) in three porous sandstones. The effective mean stresses were sufficiently high that the samples fail,~d by cataclastic flow, with development of strain hardening and shear-enhanced compaction. Comparison of the new data with triaxial compression data from a previous study shows that the critical stress states for the onset of shear-enhanced compaction are comparable for the two different loading paths. The initial yield stress data for each sandstone map out art approximately elliptic envelope in the stress space. Stress-induced permeability anisotropy was inferred from synthesis of the triaxial compression and extension data. Before the onset of shear-enhanced compaction, permeability and poro:~ity reduction are primarily controlled by the effective mean stress and stress-induced anisotropy is negligible. With the onset of shear-enhanced compaction and development of cataclastic flow, coupling of the deviatoric and hydrostatic stresses induces considerable permeability and porosity reduction. The permeability for flow along the direction of the maximum (compressive) principal stress is greater than that along the minimum principal stress. Microstructural observations on the shear-compacted samples show appreciable increase of grain crushing and pore collapse, which explain the overall decrease in permeability. The damage from grain crushing is highly anisotropic, with the stress-induced microcracks preferentially aligned with the maximum principal stress direction. Because more microcrack conduits are available to focus the flow in this direction, the permeability is relatively enhanced.
Keywords: Triaxial extersion; Cataclastic flow; Shear-enhanced compaction; Loading path; Porosity reduction; Permeability anisotropy; Stress-induced compaetio:a
1. In t roduc t ion
A fundamental understanding of the effect of stress on porosity arLd permeability is of importance in many rock mechanics applications in tectono-
* Corresponding author. Tel.: + 1-516-6328302; fax: + 1-516- 6328240.
physics and reservoir engineering. Experimental measurements under crustal conditions of pressure, temperature and stress provide useful insight into these problems. The effect of hydrostatic loading has been studied comprehensively in the laboratory. In general, increasing confining pressure (or decreasing pore pressure) would result in microcrack closure and possibly pore collapse, which are manifested by concomitant decreases in porosity and permeability
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200 W. Zhu et al. / Mechanics o f Materials 25 (1997) 199-214
(e.g. Brace, 1978; David et al., 1994). The effect of nonhydrostatic loading is more complicated, espe- cially in rocks with porosities greater than 5% or so. Results from triaxial compression tests on such porous rocks (including limestone, sandstone and shale) show that the pore space may dilate or com- pact in response to the deviatoric stress field (e.g. Paterson, 1978). When deformed under relatively low pressures, the samples will typically dilate and fail by shear localization and brittle faulting. When deformed under relatively high pressures, significant collapse of the porosity may be induced by the coupled effects of the hydrostatic and deviatoric stresses. This phenomenon of 'shear-enhanced com- paction' (Curran and Carroll, 1979) leads to strain hardening and macroscopically ductile failure, in- volving homogeneously distributed microcracking ('cataclastic flow'). Acoustic emission (Wong et al., 1992) and microstructural (Men~ndez et al., 1996) observations on porous sandstone show that the on- set and development of shear-enhanced compaction are associated with grain crushing and pore collapse. Significant reduction in permeability has also been observed (Zh u and Wong, 1997).
Permeability is intimately related to the geometric complexity of the pore space. While porosity is a scalar measure of the pore geometry, permeability is a second-rank tensor. Hence permeability is expected to be anisotropic in a rock subjected to an anisotropic stress field, especially when anisotropic damage is accumulated. Although permeability anisotropy is widely assumed in fault models (e.g. Rice, 1992; Hickman et al., 1995) and reservoir evaluation (e.g. Rhett and Teufel, 1992), there is a paucity o f experi- mental data on this tensor quantity. If the permeabil- ity and stress tensors are coaxial, then a plausible approach to characterize the permeability anisotropy in a stressed sample is to measure the flow parallel to each of the principal stresses. Unfortunately, it is very difficult in a high~pressure environment to mea- sure fluid transport in paths other than along the axial direction. Most of the available data are from conventional triaxial compression tests, in which the axial flow provides information on the permeability k t along the direction of the maximum ~principal (compressive) stress 0"t. In a seminal study, Zoback and Byerlee (1976) conducted both triaxial compres- sion and extension tests to characterize the perme-
ability anisotropy of Ottawa sand. In a triaxial exten- sion test, the minimum principal stress 0" 3 is applied axially, while the radial stress corresponds to the maximum and intermediate principal stress (0"1 = 0"2). The axial flow in this configuration provides information on the permeability k 3 along the direc- tion of the minimum principal stress (Fig. 1). This approach was adopted by Bruno (1994), who re- ported significant permeability anisotropy in three weakly cemented sandstones deformed in the brittle faulting regime. Although porosity change was not measured, his numerical simulation indicates signifi- cant dilatancy.
In this laboratory study, we focus on the evolution of porosity and permeability in porous rocks under- going shear-enhanced compaction and cataclastic flow. We conducted a series of triaxial extension experiments on three sandstones, the hydromechani- cal properties of which were previously character- ized by us under triaxial compression (Zhu and Wong, 1997). Data on permeability as a function of porosity for samples stressed in the two different configurations provide estimates on the permeability anisotropy. Furthermore, the mechanical data pro- vide insights into the effect of loading path on the compactive yield and strain hardening behavior.
2. Experimental procedure
The Adamswiller, Berea and Rothbach sandstones have nominal porosities ~ o = 2 3 , 21 and 19.9%, respectively. Petrophysical description of the sand- stones was provided by Wong et al. (!997). The Berea and Rothbach sandstone samples were cored perpendicular to the bedding, whereas the Adamswiller sandstone samples were cored parallel to the bedding. All the samples were ground to a cylindrical shape, with a diameter of 18.4 mm and a length 38.1 mm. The tolerance of the samples is 0.02 mm. Before each test, the sample was saturated with distilled water and jacketed with copper foil, and then it was positioned between two steel end-plugs, each of which has a concentric hole at the center for fluid access to the upstream or downstream pore pressure system. Heat-shrink polyolefine tubings were used to separate the sample from confining pressure medium (kerosene).
W. Zhu et al. / Mechanics o f Materials 25 (1997) 199-214 201
The samples were then deformed at room temper- ature inside a pressure vessel. The axial stress was applied by a servo-controlled hydraulic system, while the lateral stresses were applied by an air-driven pump. The pore pressure was fixed at Pp = 10 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 trans- ducer (DCDT). The porosity change was calculated from the ratio of the pore volume change to the initial bulk volume of the sample, with an uncer- tainty 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 ves- sel with a DCDT mounted between the moving piston and the fixed upper platen. The uncertainty of the axial displacement measurement was 10 ftm.
In each test, the irdtial permeability was measured at the confining pressure Pc = 13 MPa (correspond-
ing to an effective pressure of P c - P p = 3 MPa which is sufficient to load the jacket tightly around the sample, thus inhibiting leakage along the sample-jacket interface). In a triaxial extension test, the sample is first loaded hydrostatically (with 0-1 = °'2 = 0-3 = Pc) and then while maintaining the axial stress (0-3) constant, the radial stresses (0-1 = 0-2) were increased incrementally (Fig. 1). In a triaxial compression test, the sample was first loaded hydro- statically to a desired confining pressure Pc, and then the axial stress (0-1) was increased incremen- tally while maintaining the radial stresses (0-3 = 0-2 = Pc) constant. At different stages of deformation, the axial loading ram was locked and the in situ permeability was measured as a function of the stress state.
In a typical test the permeability value would change by 3 to 5 orders of magnitude. We have modified BernabCs (1987) design of a wide-range permeameter (originally for hydrostatic loading) for
pipettei l 6 metedngvalve
accumulator volumometer
: Heise gauge 2® @ 3®
8
differential I--I pore pressure ~tranducer
test gauge --1
T
pressure vessel
@ ~ outlet
hand pump 1 F ~,absolute 7 11 H pore pressure
ansducer ~ ) valve
rupture disk
/ /
triaxial compression upstream
G3
downstream pp- App i
triaxial extension upstream ~ i Pp
-ii d o w r l s ~ pp- APp
t- .o dl,,=l o .=
. m
3= O i
Fig. 1. Schematic diagram of a wide-range permeameter. The steady-state flow or pulse transient technique can be used on a sample loaded in the conventional triaxial compression and extension configurations.
202
Table 1
Exper imenta l results
W. Zhu et al. / Mechanics o f Materials 25 (1997) 199-214
Effect ive min imum
principal stress
(o" 3 - Pp) (MPa)
Differential stress Porosi ty Axial strain
(o.l - °'3) (MPa) qb (%) e 3 (%)
Permeabi l i ty k 3 ( × 10 -15 m 2)
Adamswi l l e r
3 0.00 23.00 57.80 a
10 0.00 22.20 34.50 a
20 0,00 21.60 23.00 a
30 0.00 21.28 20.60 a
40 0.00 21 .02 16.20 a
60 0.00 20 .64 12.90 a
80 0.00 20.31 0.00 10.50 a
80 57.96 20.00 - 0 . 1 8 14,70 a
80 88.04 19.75 - 0 . 3 4 11.20 a
80 126.82 19.49 - 0 . 5 4 8.00 a
80 149.32 18.66 - 0 . 7 5 3.61 a
80 167.17 14.49 - 3.23 0.62 a
80 228.53 11.50 - 15.63 0.05 b
3 0.00 23.00 68.00 a
10 0.00 22 .24 49.80 a
20 0.00 21.68 31.70 a
30 0.00 21.43 24.50 a
50 0,00 21.06 22.30 a
70 0.00 20.75 17.00 a
90 0;00 20.45 13.10 a
110 0.00 20.19 0.00 11.60 a
110 56.81 19.90 - 0 . 1 4 9.37 a
110 96.19 19.62 - 0 . 2 8 8.41 a
110 129.59 18.87 - 0 , 4 1 6.95 a
110 150.70 16.30 - 0 . 6 1 1.77 a
110 185.37 13.75 - 1.91 0.38 a,b
110 224.56 12.40 - 3.42 0.10 b
110 281.32 10.24 -- 8.33 0 .04 b
3 0.00 23 .00 66.80 a
10 0.00 22.43 51.40 a
20 0.00 22.01 46.60 a
35 0.00 21.61 34.00 a
50 0.00 21,38 26.00 a
70 0,00 21 .10 23.50 a
90 0.00 20.83 20.10 a
120 0.00 20 .36 13.30 a
150 0.00 20 .00 0.00 11.90 a
150 57.09 19.75 - - 0 . 1 4 9,68 a
150 114.40 19.09 - - 0 . 3 9 7.05 a
150 135.95 17.16 - - 0 . 5 9 2.47 a 150 148.60 15.25 - - 0 . 8 2 0.67 a
150 179.37 14.08 -- 1.11 0 .22 a.b 150 207.81 13.02 -- 1.68 0.15 a,b
150 243.55 12.01 -- 2.79 0.10 b
Berea
3 0.00 21 .00 231.00 a
10 0.00 20.61 189.00 a
20 0.00 20 .39 182.00 a
W. Zhu et al . / Mechanics of Materials 25 (1997) 199-214 203
Table 1 (cont inued)
Effective min imum
principal stress
( t r 3 - Pp) (MPa)
Differential stress Porosity Axial strain
(0" l -- 0"3) ( M P a ) q5 (%) e 3 (%)
Permeabi l i ty k 3 ( x 10 -15 m 2)
n e r e a
40
70
100
100
100
100
100
100
100
3
10
2O
40 60
80
110
140
160
160
160
160
160
3
10
2O
40
6O
8O
110
140
170
200
2OO
2OO
200
2OO
2OO
2OO
200
200
2OO
3
10
2O
40 6O
8O 110
140
0.00 19.99 172.00 a
0.00 19.54 143.00 a
0.00 19.20 0.00 121.00 a
82.22 18.89 - 0 . 2 0 116.00 a
119.68 18.60 - 0 . 4 2 134.00 a
150.48 18.38 - 0 . 6 0 111.00 a
180.10 18.13 - 0 . 7 8 73 .00 a
202 .92 16.93 - 1.17 4.53 a
260.11 1 1.79 - 9.28 0.26 a.b
0.00 21.00 188.00 a
0.00 20.35 166.00 a
0.00 19.94 133.00 a
0.00 19.53 125.00 a
0.00 19.18 120.00 a
0.00 18.92 111.00 a
0 .00 18.57 95.80 a
0 .00 18.24 87.50 a
0.00 17.99 0.00 76 .70 a
83.78 17.65 -- 0.09 83.80 a
146.93 17.14 - - 0 . 2 8 7.86 a
174.88 13.70 - - 0 . 8 4 0 .99 a
235.87 11.10 -- 3.25 0 .26 a,b
0.00 21 .00 245.00 a
0 .00 20.42 21 1.00 a
0.00 20.17 196.00 a
0 .00 19.94 165.00 a
0.00 19.64 159.00 a
0 .00 19.36 156.00 a
0.00 19.01 132.00 a
0.00 18.67 1 16.00 a
0.00 18.39 97.10 a
0.00 1 8.09 0.00 94.60 a
74.08 17.84 - -0 .15 93 .10 a
1 12.72 17.61 - - 0 . 2 4 62 .70 a
162.69 16.52 - - 0 . 3 7 1.75 a
179.52 13.84 -- 0.58 0.66 a
215.26 12.16 -- 0.95 0 .52 a 240.22 1 1.43 -- 1.25 0.41 a.b
293.72 10.34 -- 1.61 0.23 a,b
336.72 9.35 - - 2 . 1 6 0.18 a,b
392 .04 8.67 -- 2.96 O.l 2 ~b
0.00 21.00 237.00 a
0.00 20.57 208 .00 a
0.00 20.15 165.00 a
0.00 19.67 151.00 a 0 .00 19.34 126.00 a
0.00 19.02 1 19.00 a
0.00 18.54 108.00 a
0.00 18.24 98 .10 a
204
Table 1 (continued)
W. Zhu et al. / Mechanics o f Materials 25 (1997) 199-214
Effective min imum
principal stress
(o- 3 - Pp) (MPa)
Differential sa'ess Porosi ty Axial strain
( t r I -- or3) (MPa) q5 (%) ~3 (%)
Permeabi l i ty k 3 ( )<10 -15 m 2)
Berea 170 0.00 17.92 90 .20 a
200 0.00 17.64 80.20 a
230 0.00 17.35 63 ,60 a
250 0.00 17.16 0.00 54 ,30 a
250 79.89 16.78 - - 0 . 0 2 31 .80 a
250 138.85 14.61 - -0 .10 0.72 a
250 155.23 13.51 - -0 .17 0 .64 a
250 194.41 11.88 - 0 . 2 9 0.41 a,b
250 252.08 10.74 - 0 . 8 1 0.27 a.b
Ro thbach
3 0.00 19.90 129.00 a
10 0.00 19.24 108.00 a
20 0.00 18.69 83 .80 a
30 0.00 18.35 69 .50 a
50 0.00 17.80 47 .00 a
70 0.00 17.36 40 .50 a
90 0.00 16.95 0.00 33 .80 a
90 48.23 16.57 - 0 . 1 7 28.60 a
90 83.18 16.23 - 0 . 3 0 26 .40 a
90 104.51 15.59 - 0 . 4 2 8.50 a
90 119.84 13.15 - 1.31 0 .60 a
90 160.29 9 .84 - 4 . 9 3 0.10 b
90 213.93 8.27 - 10.07 0.04 b
3 0.00 19.90 153.00 a
6 0.00 19.63 131.00 a
10 0.00 19.27 81.20 a
20 0.00 18.76 54 .50 a
30 0.00 18.42 42 .50 a
40 0.00 18.19 36.80 a
60 0.00 17.77 26.80 a
80 0.00 17.43 19.50 a
100 0.00 17.09 14.10 a
126 0.00 16.62 0.00 10.50 a
126 18.47 16.28 - - 0 . 0 2 7.09 a
126 62.23 15.95 - -0 .03 4.07 a
126 86.17 15.51 - 0 . 1 3 2.02 a
126 117.32 14.09 - 0.33 0.29 a,b
126 163.83 11.48 - 1.93 0 .10 b
126 210.26 9.98 - -4 .08 0.05 b
3 0.00 19.90 191.00 a
8 0.00 19.38 102.00 a
16 0.00 18.87 75 .60 a
30 0.00 18.35 48 .80 a
50 0.00 18.03 28 .30 a
70 0 .00 17.61 15.90 a 100 0.00 17.08 6.34 a
130 0.00 16.58 3.46 a
165 0.00 16.09 0.00 1.88 a
165 78.62 15.71 - -0 .21 1.36 a
W. Zhu et al. / Mechanics o f Materials 25 (1997) 199-214 205
Table 1 (continued)
Effective minimum principal stress
(°'3 - Pp) (MVa)
Differential stress Porosity Axial strain
(0-1 -- 0"3) ( M P a ) ~b (%) e 3 (%)
Permeability k 3 ( × 10 -15 m 2)
Rothbach 165 165 165 165 165
126.46 15.16 - 0 . 3 3 0.47 a,b 151.48 13,33 - 0.80 0.11 a.b 222,30 11.07 - 2.75 0.04 b 267,27 10.22 -- 3.85 0.03 b 275.90 9.19 -- 6,81 0.01 b
a Values obtained by using the steady state flow technique. b values obtained by usirLg the pulse transient technique,
operation in a triaxial system. Details of the experi- mental procedure were described by Zhu and Wong (1997). Two different techniques can be imple- mented, depending on the permeability of interest. If the permeability at a certain stress state is relatively high ( k > 10 -16 m~), it is measured by the steady state flow technique. For k < 10 -16 m 2, thermal fluctuation would render it difficult to achieve steady state flow, and instead we use the pulse transient technique first described by Brace et al. (1968). A small pore pressure difference A P of less than 1 MPa was used in both techniques. Typically, the duration of permeability measurements was 10 min to 1 h. Hsieh et al. (1981) have shown that the exact solution for the pressure transient is sensitively de- pendent on the storage characteristics. In our setup, the compressive storage of the upstream reservoir is 3.18 × 10 -8 ma/MPa. The downstream reservoir includes an accumulator, and consequently its stor- age is effectively infinite. These storage values fall within the parameter space in which the transient can be approximated as an exponential decay (Neuzil et al., 1981; Bernab6, 1987). Accordingly, the perme- ability can be inferred from the slope of a semilog plot of the pressure difference (between upstream and downstream) versus time, following the proce- dure pioneered by Brace et al. (1968). For all of our tests, the uncertainty of permeability measurements is < 15%.
3. Experimental results
We have compiled in Table 1 all our permeability data as functions of stresses, axial strain and poros-
ity. We will adopt the convention that compressive stresses and compactive strains (i.e. shortening and porosity decrease) are positive. The differential stress (Q = oq - 0- 3) and permeability (k 3) are plotted as functions of the axial extension for the (a) Adamswiller, (b) Berea, and (c) Rothbach sandstones in Fig. 2. The effective minimum principal stress (0-3 - P p ) for each experiment is indicated. The per- meability and porosity are also plotted as functions of the effective mean stress (P = (20" l + 0-3)/3 - Pp) (Fig. 3). For reference, the evolution of perme- ability and porosity under hydrostatic loading are also included as dashed curves.
Several features of the mechanical data should be noted. First, all the samples show appreciable axial extension (Fig. 2). For a given sandstone, the exten- sion ( - ~ 3 ) decreased with increasing 0- 3. Second, the differential stress increased with deformation, and strain hardening inhibited the development of shear localization. The typical failure mode is shown in Fig. 4b for the Berea sandstone sample deformed at 0- 3 - Pp = 100 MPa (Fig. 2b). The axial extension of ~ 9% was homogeneously distributed and there is no sign of shear localization or necking instability. Third, porosity change in a triaxially extended sam- ple coincided with the hydrostat up to a critical stress state (indicated by C* in Fig. 3), beyond which there was an accelerated decrease in porosity in comparison to the hydrostat. At stress levels beyond C*, the deviatoric stress field provided significant contribution to compaction, and accordingly this phenomenon is called 'shear-enhanced compaction' (Curran and Carroll, 1979; Wong et al., 1992).
All our samples show decrease of permeability with deformation. Since it is sensitive to subtle
206 W. Zhu et al./ Mechanics of Materials 25 (1997) 199-214
a)
ill.
. B e -
"o
E
¢n
6~
300
200
100
0
1 0 - 1 4
10-15
10-16
10-17
AdamswUler sandstone (4 =23 %)
0 2 4 6 8 10 12 14 16
~ 80MPa
b)
400
a .
300 ¢1 Itl
• ~ zoo
t~ o ~
e- 100
"o 0
10-13
E ~ 10_14
,,Q
~ 10_15 L.
Q,
10-16 -
~ Berea sandstone (qb o =21%)
fPa ~200MPa
_ / 160MPa
a x i a l extension (%) , I , I , I , I , I
0 2 4 6 8 10
250MPa ~ 200MPa
300 t~ O.
01 200
m m :~ 100
~ o
E
e~ t~
e,
10-14
10-15
1 0 - 1 6
lO-l~
Rothbach sandstone (~ =19.9%)
~ 165MPa
Y 126MPa
10
90MPa
Fig. 2. Experimental data for (a) Adamswiller, (b) Berea, and (c) Rothbach sandstones. The differential stress (o" I - 0- 3) and permeability k 3 are plotted versus axial (extensive) strain. The effective minimum principal stress (0-3 - Pp) for each experiment is as indicated.
Fig. 3. Porosity and permeability as functions of effective mean stress for (a) Adamswiller, (b) Berea, and (c) Rothbach sandstones. The solid symbols represent data obtained at triaxial extension stress states. The open symbol represent data obtained at hydrostatic stress states. The critical stress states C * for the onset of shear-enhanced compaction are indicated by the arrows, and 0" 3 - Pp for each experiment is as indicated. For reference the hydrostatic compression data are shown as dashed curves.
W. Zhu et a l . / Mechanics o f Materials 25 (1997) 199-214 207
a) 24
~ 20
._~ is
0 16 Q.
8
10-13
¢q
. ,~ 10 -t4
.,0 CO O)
~ 10.1 s D.
10"16
10-1"~
Adamswiller sandstone (conventional triaxial extension tests) b) 23
C*C*c.
.~- 13 ",oh ydr°static
80MPa \ ~ ~V " '" 0 e'.
"llO~OMP2 50M~a " ' " ' " o 8
[ t I t I t I i I 10-]2
O0 3OO 1oo effective mean stress (Ml~a °°)
~ lO_t 3
10.] 4
I:. "" o hydrosta, fic
15~M;a" ...... lO'~S
110MPa
10"16
Berea sandstone (conventional triaxial extension tests) ~ C* 7 .C.
ol
100MP¢ ~ ! i st~'°
I i I i I E I i I i I i I 0 100 200 300 400 500 600
effective mean stress (MPa)
"o.
l OOMPa 200MPa
c) 2O
18
16
A
t4 m e 12 0 el
10
lO-t3
• ~ lo-sd m
0 I~ 10 "15
10"16
10-17
Rothbach sandstone (conventional triaxial extension tests)
, C *
~"6A ~'-fk
90MPa f I i I
100 200 300 400
208 W. Zhu et al. / Mechanics o f Materials 25 (1997) 199:214
4. Discussion
a) b) e) d) e)
Fig. 4. Failure modes of Berea sandstone samples: (a) shear localization and (b) cataclastic flow under lriaxial extension; (d) shear localization and (e) cataclastic flow under triaxial compres- sion. For reference an undeformed sample (c) is also shown. The sample (b) has undergone a relatively homogeneous axial exten- sion of ~ 9% without development of geometric instability. The bulge-shaped sample (e) was subjected to an axial shortening of ~ 30%,
differences in the pore structure from sample to sample, the permeabili ty usually shows more scatter. Permeabili ty and porosity changes tracked one an- other, with the most significant decreases in samples deformed to beyond the compactive yield stress C *. The relatively good correlation between permeabili ty and porosity changes observed here is similar to our triaxial compression results (Zhu and Wong, 1997), although the magnitude of the permeabili ty decrease seems to be somewhat larger in the extension tests.
4.1. Critical stress for the onset of shear-enhanced compaction
Triaxial extension data for the critical stress C * (Fig. 3) show that the differential stress (Q) at the onset of shear-enhanced compaction decreases with increasing effective mean stress (P ) . Similar trends were observed in sandstones deformed in triaxial compression. Wong et al. (1997) demonstrated that the triaxial compression data for C* mapped out an elliptical yield envelope in the ( P , Q) stress space, as described by critical state soil mechanics (Scho- field and Wroth, 1968) and cap model (DiMaggio and Sandler, 1971). The experimental data may be generalized to other loading paths if the initial yield stress is assumed to be a function of only the first and second stress invariants. Indeed if this assump- tion is valid, then the compactive yield stress data for the two different loading paths should fall on the same cap. To test this, we plotted in Fig. 5 our new extension data and triaxial compression data from fluid flow experiments presented by Zhu and Wong (1997). The agreement is reasonable, suggesting that
250
I1. 200
0
150
" - - 100 C
"0 50
N ~ 0 ' ~ ' A I 0 ' ~ '
0 100 200 300 400
O
e f f e c t i v e m e a n s t r e s s , P ( M P a )
Fig. 5. The stress states C * at the onset of shear-enhanced compaction are shown in the (P, Q) stress space. The solid symbols represent triaxial extension data from this study. The open symbols represent triaxial compression and hydrostatic data of Zhu and Wong (1997). The shaded area indicates schematically the domain in which dilatancy was observed (in triaxial compression experiments). The lower and upper boundaries correspond, respectively, to onset of shear-induced dilation and peak stress in the brittle faulting regime (Wong et al., 1997).
w. Zhu et aL / Mechanics of Materials 25 (1997) 199-214 209
the compactive yield behavior is not very sensitive to loading path at these high mean stresses. However, it should be noted that this test is still limited because only axisymmetric loading is considered.
A common feature, of these experiments is that the differential stress was not applied continuously at a constant rate. In each experiment, the loading ram was locked during several intervals (each lasting for an hour or so) while permeability was measured. Some stress relaxation was usually observed during the ' locked' intervals. As a result, yield stresses of Zhu and Wong (1997) were generally lower than those obtained by Wong et al. (1997) for continuous loading in triaxial compression. The apparent weak- ening of the sample,; compared to the compression experiments was possibly due to additional damage from stress corrosion incurred during the stress re- laxation intervals.
Wong et al. (19971) show that triaxial compression data for 6 sandstones with porosities ranging from 15 to 35% map out el:liptical yield envelopes whose major and minor axes increase with decreasing porosity and grain size. Furthermore the plastic
strains at initial yield have ratios comparable to those predicted by the normality condition (Drucker, 1951). Qualitatively similar behavior was observed in car- bonate rocks (Brown and Yu, 1988), shale (Steiger and Leung, 1991), and sediments (Jones and Addis, 1986) deformed in triaxial compression. Our sand- stone data suggest that these observations on other rock types may also be generalized to triaxial exten- sion loading.
4.2. Permeability anisotropy inferred from triaxial compression and extension tests
We observed dramatic decreases of k 3 after the onset of shear-enhanced compaction (Figs. 2 and 3). In triaxial compression tests, Zhu and Wong (1997) also observed significant decrease of k I as soon as C * was exceeded. Altogether these data suggest that the compactive yield envelope defines the boundary of a closed domain in the stress space, within which both porosity and permeability decrease with increas- ing hydrostatic loading, with negligible dependence on the deviatoric stresses. When the rock is loaded to
Berea sandstone
.~, 400 2 o / ~
250MPa (c)
! 300 ~ M e , ,
"~[ 200
~i 100
0 0 100 200 300 400 500
effective mean stress, P (MPa)
Fig. 6. Loading paths of triaxial extension and compression tests. Solid lines represent triaxial extension tests with fixed values of o.3 - Pp as indicated. The symbols are stress states at which permeability measurements were conducted. The dashed line represents a triaxial compression experiment with o- 3 - Pp fixed at 250 MPa. Note that the loading paths of extension tests and compression tests have at maximum only one stress ,,;tam coinciding.
210 W. Zhu et al. / Mechanics o f Materials 25 (1997) 199-214
a)
A .g
m i m
E
0.1
0.01
0.001 8
. ~ ~ 80MPa 110MPa 150MPa
. . . . 150MPa (c)
I
10
I I
............. g v *
V
V
Adamswiller sandstone
I [ i I I I I
12 14 16 18 20 porosity (%)
b)
0.1
t~
E L 0.01
. ~ 100MPa 160MPa 200MPa 250MPa
. . . . 250MPa (c)
v
• V
WV (rl i'
t'
.V / 1
Berea sandstone 0.001
i i I I I I I
8 10 12 14 16 18 porosity (%)
Fig. 7. Normalized permeability versus porosity reduction for the (a) Adamswiller, (b) Berea, and (c) Rothbach sandstones. The solid symbols are data from extension tests at fixed 0- 3 - Pp as indicated, and the open symbols joined by dash lines are from compression tests. More pronounced reduction of permeability was observed for flow along the direction of the minimum principal stress.
W. Zhu et al . / Mechanics of Materials 25 (1997) 199-214 211
A 4g
V
i n m i n
0 E t . _
0
e)
0.1
0.01
0,,001 8
9 0 M P a
1 2 6 M P a
1 6 5 M P a
- - - 1 6 5 M P a ( c )
.4,W"
T
10
o~ ol . ~
/ / "
j 1 ~ J ' ~
Rothbach sandstone
I I , I
12 14 16 porosity (%)
Fig. 7 (continued).
beyond this cap, significant decreases of porosity and permeabilities (k I artd k 3) result from the coupled effects of the deviatoric and hydrostatic stresses.
To infer the permeability anisotropy, one would ideally like to compare the values of k~ and k 3 in samples with the s~Lme porosity subjected to the same stress state. However, since the compression and extension tests follow different paths in the (P, Q) stress space (Fig. 6), two such experiments (with the same fixed values tr 3 and Pp) would at most intersect at one point. Therefore these tests can not provide detailed information on anisotropy of sam- pies subjected to the same stress states. To circum- vent this problem, several approximate approaches have been proposed. Zoback and Byerlee (1976) assumed that the permeabilities were primarily influ- enced by the axial strain. They calculated the perme- ability anisotropy of Ottawa sand as a function of the effective minimum principal stress 0" 3 - - ep for fixed values of the principal strains e, ( = 10%) and s 3 (= - 3%), with corresponding permeabilities k I and k 3 determined in triaxial compression and extension tests, respectively. Since each permeability ratio was
calculated from measurements on two samples with different porosities and stress states, these estimates of permeability anisotropy involve significant uncer- tainty. Bruno (1994) compared the data for a single sample which was subjected to a loading cycle in triaxial compression before it was loaded in triaxial extension. If irreversible strain was accumulated in the first loading cycle, then the pore structure of the sample was different in the extension test.
A somewhat different approach is adopted here. Our data indicate that the anisotropy in permeability is negligible before the onset of shear-enhanced compaction. Since porosity is a first-order measure of the pore geometry, we will use it as the indepen- dent variable for the inference of permeability ani- sotropy. Because the permeability is very sensitive to slight variation in the initial pore structure from sample to sample, we normalized the data with respect to the permeabilities immediately before the onset of shear-enhanced compaction. For a given sandstone, the compactive yield stresses C* were attained at approximately the same porosity (Fig. 3), which will be denoted by ~b *. Specifically, we chose
212 W. Zhu et a l . / Mechanics o f Materials 25 (1997) 199-214
th * = 20, 18 and 16% for the Adamswiller, Berea and Rothbach sandstones, respectively. For each ex- periment, the specific permeability value at this porosity was denoted by k*, and the permeability data beyond the onset of shear-enhanced compaction were normalized to arrive at the ratios k3/k *. The normalized permeabilities are plotted as dark sym- bols in Fig. 7. Triaxial compression data of Zhu and Wong (1997) were similarly normalized, and the ratios kJk* (open symbols joined by dashed lines) were plotted as function of porosity in Fig. 7.
The normalized data indicate that shear-enhanced compaction reduces the permeability more effec- tively along the direction of minimum compression. As a result, the anisotropy in samples stressed to beyond C* is such that k~ > k 3. The permeability anisotropy is very pronounced in the Adamswiller
and Berea sandstones. In comparison, the Rothbach sandstone (with the lower initial porosity) shows the weakest anisotropy.
The anisotropic evolution of permeability seems to correlate with damage accumulation. In Berea sandstone samples deformed to beyond C*, Men6ndez et al. (1996) observed pervasive damage in the form of microcracks emanating from grain contacts. The stress-induced cracking would lead to grain crushing and pore collapse, as illustrated by the micrograph shown in Fig. 8. The pore collapse re- sults in an overall reduction of permeability for flows in all directions. However, significant anisotro- py in microcracking was evident in the deformed samples. The density of microcracks aligned parallel to o-~ was about twice that of cracks parallel to tr 3. The preferentially aligned microcracks probably pro-
03
Fig. 8. Scanning electron micrograph showing anisotropic microcracking in a Berea sandstone sample. This sample was shear-compacted under triaxial compression up to 0-t - 0- 3 = 585 MPa at constant 0" 3 - Pp = 250 MPa. The max imum and min imum principal stresses were along the vertical and horizontal directions, respectively. Pore collapse and microeraek density were characterized quantitatively by Men6ndez et al. (1996).
W. Zhu et aL /Mechanics of Materials 25 (1997) 199-214 213
vide additional conduits for flow along the 001 direc- tion, thus relatively enhancing the permeability. A similar conclusion seems also to apply to unconsoli- dated materials, since Zoback and Byerlee (1976) also observed k I > k 3 in Ottawa sandstone samples after the onset of grain crushing.
Our data also suggest that the permeability anisot- ropy increases with decreasing 0-3- However, this apparent trend is difficult to confirm because the stress states of the samples were quite different. Alternative loading paths can be adopted so that permeability anisotropy can be characterized on sam- ples with the same porosity subjected to the same stress state. Presumably this involves modifying the standard loading procedure of keeping o-3 fixed. For o- l along the axial ,direction, we may increase the axial and radial stresses proportionately so that the loading path of this modified compression test coin- cides with that of the conventional extension test (with 0- 3 fixed as indicated in Fig. 6). Comparison of data from such a modified compression test with the conventional extension data may provide better constraints on the pe:rmeability anisotropy.
In this study, we have focused on the cataclastic flow regime. The logical next step is to investigate the effect of stress on permeability and its anisotropy in the brittle fracture regime. Zhu and Wong (1996, 1997) concluded that there are two different types of correlation between permeability (along the maxi- mum compression direction) and porosity change in rocks which show dilatancy and fail by shear local- ization. In relatively porous rocks, permeability would decrease even if the deviatoric stresses induce the rock to dilate. Such a negative correlation be- tween permeability and porosity changes was also observed by Bruno (1994) in three sandstones with tho between 25 and 30% for two flow directions (parallel to 001 and 0-3). His measurements suggest that k~ > k 3. In comrast, a positive correlation be- tween permeability and porosity changes is generally observed in rocks of relatively low porosity (~b o < 15% or so). Such stress-induced enhancement of permeability in a dilating sample has been observed in Westerly granite (Zoback and Byerlee, 1975) and Darley Dale sandstone (Mordecai and Morris, 1971; Zhu and Wong, 1997). However, the measurements were for flow along the 00! direction. Since the damage is primarily in the form of anisotropic mi-
crocracking, it is plausible that the permeability is also anisotropic with k~ > k 3. However, such a con- jecture has to be validated by future experimental investigation, which will also provide important data on the magnitude of the anisotropy.
5. Summary
Laboratory measurements of porosity and perme- ability reduction under triaxial extension have been presented in this paper. The permeability was mea- sured along the axial direction. Three different sand- stones with high initial porosities were used. All experiments were conducted at room temperature. The range of effective pressure was from 80 to 250 MPa, and the pore pressure was fixed at 10 MPa. Under these circumstances all the tested samples failed by cataclastic flow. The deformation and fail- ure mode of the samples were homogeneous, without shear localization or visible necking.
The evolution of physical properties of the sam- ples showed some similarities with our previous experimental results under triaxial compression (Zhu and Wong, 1997). Specifically, the data follow the hydrostatic experiments until a critical stress state C*, beyond which significant porosity reduction, and drastic permeability drop up to 3 orders of magnitude occur. The evolution of permeability dur- ing cataclastic flow mimics that of porosity changes. The values of C* in the triaxial extension tests are also consistent with those from triaxial compression tests. This implies that C* is mainly controlled by the first and second stress invariants, and insensitive to the loading path.
Anisotropy of the permeability tensor was in- ferred from comparison of the permeability values in triaxiat extension tests to those under triaxial com- pression. We conclude that the permeability tensor component k I (in the direction of the maximum principal stress 0-1) is generally higher than k 3 (in the direction of the minimum principal stress 003). The phenomenon can be qualitatively explained by stress-induced microcracks which developed prefer- entially in orientations parallel to 001, providing addi- tional conduits for flow along the 001 direction. The magnitude of stress-induced permeability anisotropy varies with rock type as well as experimental condi- tions.
214 W. Zhu et aL / Mechanics of Materials 25 (1997) 199-214
Acknowledgements
We have benefitted from discussions with Yves Bernab~, Christian David, Joanne Fredrich, Beatriz Men6ndez and Bernard Seront during various stages of this project. We thank the anonymous reviewers for their constructive criticisms. The second author's exchange visit to Stony Brook was sponsored by the Magist~re Interuniversitaire de Physique at the Ecole Normale Sup~rieure, Paris. This research was sup- ported by the National Science Foundation under grant EAR9508044.
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