Inl J Rock Mech Mm S c t & Geomech Abstr Vol 22, No 3. pp 163-171. 1985
Printed m Great Britain All rights reserved
0148-9062/85 $3 0 0 + 0 0 0 Copyright g 1985 Pergamon Press Ltd
A Theoretical Analysis of Tectonic Stress Relief During Overcoring T-F WONG* J B WALSHt
The direct stram-gauge techmque for measurmg tectomc stress ts analy:ed theorettcally The effect of overcormg Is stmulated by imposing umform displacement dtscontmuttles rather than untform stress rehef on the cyhndrtcal surface. This stmphficatton allows algebraic expressions to be found gwmg stram at the rosette m terms of depth of overcormg These e.xpresstons are used to find Polsson's ratto and tectomc stress from data at two sttes m New York State.
INTRODUCTION The state of stress In the earth is a basic parameter reqmred for virtually all studies of geophysical processes, and, as a result, considerable effort has been expended in making measurements of stress and refining the interpretation of them Direct measurements of stress are restricted, for practical reasons, to shallow depths m the earth, and measurements below several kilometers are u n c o m m o n Here, we consider measurements made right at the earth's surface Such measurements are relatively cheap and relatively easy to carry out. They have the disadvantage that surface outcrops are apt to be fractured and weathered, and so sites are restricted to regions where hard, competent rocks are exposed. Such sites are not uncommon, and high quality measurements of surface stresses have been made by a number of investigators (for recent reviews of stress measurement, see [1,2]) Stresses are found to change with depth, but the o n e n t a u o n s o f p r m o p a l stress directions found from measurements at or near the surface seem to correlate closely with tectomc features caused presumably by stresses kdometers or tens of kdometers below the surface (see, e.g. [3-5]) Apparently the outcrop, m some cases at least, is mechanically coupled to rock at depth, and surface measurements follow the pattern imposed by tectonic stresses on the formation as a whole The type of measurement with which our analysis is primarily concerned is called the direct strata-gauge technique A strata-gauge rosette is bonded to an exposed rock surface Then the rock to which the rosette is attached is isolated from the surrounding formation by overcormg around the gauge with a core-drill Tec-
* Department of Earth and Space Soences, SUNY, Stony Brook, NY 11794, U S A tDepartment of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
tonic stresses acting on the rock cylinder are reheved as the core-drill advances, until eventually the strain reaches the completely relieved value. Typical plots of strain as function of coring depth are shown m Figs 10a and I la, note that strain mcreases with increasing core depth, reaches a m a x i m u m value, and then decreases asymptotically to the fully-relieved value Tectonic stresses rather than strains are of interest in most studies. Stresses typically are computed from the asymptotic values of strain using appropriate equations derived from elastic theory for lsotroplc, homogeneous, linearly-elastic material Values of elasuc modulus and Poisson's ratio needed in the equations are found from deformation measurements on the rock core In the laboratory. One would expect that the elastic properties of the formauon, as well as the remote stresses, determine the shape of curves gwmg strata change as a function of overcorlng depth Here we investigate the possibility that field measurements such as those given in Figs 10a and l la can be used to provide Information about the properties of the formation. To study this problem we analyze the process using the simplest model which retains the basic elements of the problem. The site IS assumed to be a half-space which is lsotroplc, homogeneous and hnearly elastic, as in previous studies. In the actual overcorlng operation, the tectonic stress acting on the overcored rock IS reduced to zero We simulate this operation by introducing an orthogonal set of point-force dipoles In the half-space over the overcored surface (see Figs 2 and 3) The analysis is approximate in that the core and the surrounding medium are subjected to a uniform change in radial displacement on the boundary of the overcore rather than a uniform change m normal stress This approxlmaUon Is a very good one for long cores in regions away from the attached end because the stress and displacements there are nearly uniform The approximation is less satisfactory for short cores near the free end. The approximation is found to be adequate, nevertheless, as we d~scuss later
163
164
WONG and WALSH TECTONICSTRESS RELIEF DURING OVERCORING 0" 2
o" 1
it X
4
I
/ o" 2
Fig 1 Co-ordmatesystem for the overconngoperauon Remotestresses o~ and o~ are assumed to be compresswewnh ot > o:
ANALYSIS We dlustrate m Fig 1 the geometnc configuration of the direct strain-gauge overconng techmque. One would expect that the pnnclpal stresses near the earth's surface are normal to or parallel to the surface, and m s t t u measurements [1,2] have verified this m numerous locations The overconng operauon therefore is oriented normal to the surface, as m Fig. 1 The v a n a u o n with depth of tectomc stress can be assumed to be neghglble over the rather short core lengths used m the &rect strata-gauge techmque The problem, then, is to determine the magmtude and &rectlon of two horizontal principal stresses a~ and o2 (where here compression is posmve and a~ > a_,) reheved by the overcormg. In a typical case, a~ and a2 are compresswe, and so both the core and the surrounding rock near the raterface expand as stress is reheved The strata field could be found by calculating &splacements m a half-space wlth an annular cut loaded by a umform tensile stress This problem must be solved using numerical techmques such as fimte element or boundary integral element methods This procedure has the &sadvantage that extenswe numencal computaUon is reqmred to cover the range of elastic properties, overcormg depth, and remote stress states tn which we are mterested We chose to analyze this problem using a simpler model which retains the basic elements of the problem and has the advantage that solutions can be obtained in closed form First, note that the displacements of the surfaces of the core and the adjacent rock due to rehewng stresses can be thought of as a &splacement &scontmmty across a cyhndncal surface m a half-space The magmtude of the &scontmmty vanes with position such that the stress change on the core surface equals the reheved stresses am and a_~ Following the approach used
to analyze surface displacements induced by faulting, we assuame here that the displacement &scontinulty does not vary with &stance along the core axis The magmtude of the displacement discontinuity is estabhshed following two schemes For the first level of approxlmauon, we assume that the &splacement &scontmmty does not vary with overcormg depth, and so the magmtude is equal to the value for a very long core, which can be calculated using simple elasuc theory For the next level of approximation, we calculate the stress change on the core at the free surface and adjust the magmtude of the displacement so that the stress change equals the remote stress state Both schemes produce approximate solutions because the stress relieved by overcormg is not uniform over the core surface m these models As we show, however, the approximation is sufficiently accurate for the present application where field measurements themselves are not precise The medium is assumed to be homogeneous, Isotroplc and hnearly elastic, with Young's modulus E and Polsson's ratlo v The co-ordinate system zs shown m Fig I, where the x- and )'-axes are parallel to am and a., respectwely The z-axis coincides with the core axis, posmve downwards We are interested m calculating stratus at the o n g m (0, 0, 0), l e on the surface at the centre hne of the core The general problem where a~ :~ o_, is separated into two simpler cases one wlth equal remote stresses trm=a2=o0, and the other with remote shear, l e a] = - a 2 = z0 In the first case, the &splacement dlscontlnmty has only a radial component because of the symmetry of the remote stress field and core configuration For equal blaxml remote stresses, Weaver [7] and Brady and Bray [8] show that a ra&al displacement &scontlnmty is eqmvalent to the orthogonal set of tensile &poles illustrated m Ftg 2 Note that the dipoles
WONG and WALSH TECTONIC STRESS RELIEF DURING OVERCORING
165
0 o
x1
I I
y)
g
/
i I I
I
,
1
I I
I ) ) I
z~
I I I
Z Fig 2 Rehewngnormal stress at a point at depth c on the overcored surfaces ~s simulated by ~mposmg a radml displacement dtscontmmty conslstmg of three dipoles (fl,f:,fO where f, --f3 = vfd(l - v) Both the (x, y, c) and (x',)', =') co-ordinate systems m the fllustratmn are used m the calculatmns descnbed m the Appendix
are n o t equal m m a g n i t u d e the aztmuthal a n d axml c o m p o n e n t s both have m a g n t t u d e s which are v/(1 - v) times that of the radtal c o m p o n e n t The dtsplacement field resulting from the d ] s t n b u t m n o f dipoles o n the cored surface lS f o u n d usmg M m d h n ' s [9] c o m p i l a t i o n of solutions for pomt-force sources m a semt-mfinlte space The analysis mvolves extenswe algebratc calculations, a n d so we gtve only an o u t h n e o f the steps mvolved m A p p e n d t x I We find that the strain at the origin for a core o f radius a a n d depth h can be expressed in a simple algebrmc form, If the d l s l o c a t m n ts assumed to be u n i f o r m with c o r m g depth, the expression for strain is e ~ = e,, = (ao/E)
)
(,_v) ~
Ftg 3 Rehevmgshear stress at a point at depth c on the overcored surfaces ts s]mulated by tmposmg a double couple g m a plane normal to : The (x. y, x) and (x',y'. :') co-ordinate systems are referred to in the Appendix
a n d E ( k ) a n d K ( k ) are elhptlc mtegrals [14] The stress change a ~ ( = a,,) associated with the stram change given by (1) a n d (2) ts simply a , ~ = a , , = e ~ , E / ( l - v ) = e,, E/(1 - v). The v a n a t l o n of strain, stress a n d dipole strength with depth of overcormg are plotted as a f u n c t t o n o f P o l s s o n ' s ratto In Figs 4, 5 a n d 6 N o t e that stress a n d strain reach a peak before d e c a y m g to the asymptottc value at large overcormg depths, this behavlour ts stmtlar to that observed m field m e a s u r e m e n t s such as those gwen m Figs 10a a n d 1 l a N o t e also that the peak value o f stress a n d stram depends o n Po~sson's ratio This d e p e n d e n c e ts s h o w n more clearly m F~g. 7,
[2(2 - v ) ( h / d ) - 6 ( h / d ) 3 + 3(h/d)5],
(1)
where d - ' = a - ' + h 2 If we assume that the displacement d l s c o n t m u t t y v a n e s wtth coring depth such that the stress change on the core at the free surface equals the remote stress ao, strata at the rosette is gtven by the expression
W
b°16 yoO c o N O8 --
e,~ = e,, = rC(ao/E)
[2(2 - v ) ( h / d ) - 6 ( h / d ) ~
20r-
04
8
z
+ 3(h/d) 5],
(2)
0
I 1 Normahzed
where I(k, v) = ~/(1 - kZ)[2(1 + k 2) E ( k ) - [4(1 - v) + k:] K(k)] k " = 4aZ/(h 2 + 4a2),
I 2
I 3
J 4
core depth ( h / a )
F , g 4 S t r a t a e,~(=~,,) a t (0, 0, 0 ) n o r m a h z e d w i t h r e s p e c t t o ao/E f o r t h e c a s e o f e q u a l b t a x l a l r e m o t e s t r e s s e s a~ = a 2 = a 0 ( s e e e q u a U o n 2)
Note that the strata reaches a maximum where the overconng depth h ts equal to approximately one-half the core radms a Note also that Polsson's raUo v has a strong effect upon strata
166
WONG and WALSH 28
26
TECTONIC STRESS RELIEF DURING OVERCORING
/
v*045 o
bO24
3
"~" Z 2
,b zo
g
'18
~
~2 o
+
~o
~ os 06
~
o4 I 1
0
I 2
Normohzed
I 4
1
3
o o+ ,.,r
I
core depth ( h / o )
Fig 5 C h a n g e in stress o,,(=<7,,) at (0, 0, O) n o r m a l i z e d with respect to tro for the case o f equal blaxlal r e m o t e stresses tr~ = tr: = ao
where we have plotted the ratio of peak strain to the asymptouc value as a function of Po~sson's ratio Calculating the strata at the strata gauge ~s more comphcated m the case of remote shear because both radml and azamuthal components of &splacement &scontmmty vary over the ctrcumference of the core The azamuthal component as eqmvalent to a c o m b m a u o n of two &poles with moment [7, 8], often referrred to as a double couple Such a double couple as shown m Fag. 3 In calculating the strata change due to rehewng the shear component of the remote stresses, we consader only the case where the dapole strength ts independent of overcormg depth As we &scuss later, thts approximation gwes results which are sufficiently accurate for the case where the remote stress state as shear Detads of the calculation are gaven m Appen&x II, where we show that the stram and stress change at the centre-hne of the core due to rehevlng the shear component z0 of the remote stress can be expressed m the following form
=
e++= --e,+
--
I 01
5
(z/E ~ (l+v) [-(5-4v)h o/ ,(]-]--~v) L -d-
6(l-2v)h d+h
I 02
I 03
Polsson
ratio (~,]
I 04
I 05
Fig 7 The ratio of p e a k strain rm,,, to fully reheved strata e+ ( ~ h e r e = ~,, = ~,, at (0, 0. 0)) as a function o f Poisson's ratio for the case where the r e m o t e stress state is equal bmxml stress an This curve provides a c o n v e n i e n t m e t h o d for e s t i m a t i n g P m s s o n ' s ratio from held m e a s u r e m e n t s o f strain as a function o f o v e r c o n n g d e p t h
Figures 8 and 9 dlustrate the varmtlon of strain and stress calculated from (3) and (4) wath depth o f overc o r m g for various values o f Pmsson's r a u o N o t e that the curves m F~gs 8 and 9 are not peaked lake the curves m F~gs 5 and 6, where the average normal stress c o m p o n e n t o f the remote stress state as constdered COMPARISON
WITH DATA
E n g e l d e r a n d S b a r [I0, 1 I] a n d P l u m b et al [5] m a d e n u m e r o u s o v e r c o r m g stress m e a s u r e m e n t s o v e r a n extensive area m the A&rondack Mountains m New York
Many of these measurements were the &rect stramgauge type because the techmque as relatwely quack and cheap Engelder furmshed us wtth two unpubhshed strata rehef records, chosen arbltrardy from those measurements whach appeared to be rehable The records
16
(3)
,
I/= 0 45 14
zo [-(5 - 4v)h (1 + 2v)L d
tr,,= - t r , ~ -
6(1 - 2v)h d + h
d'J
w v*O
(4) w '
¢q
7
6 == 4 -~
3
-6
2
y
i
08 O6
~" 02 O0 -0z
i
1 1
I Z
I 3
g E ~5 z
Normohzed o
1
2
3
4
Normohzed
5 core
6
7
8
9
10
depth ( h / o )
core
deplh ( h / o )
Fig 8 Strata ~,,( = -~,,) at (0,0,0) normalized with respect to z./E for the case where the tectomc stress state is pure shear (oj = - 0 2 = z,,) Note that the relation between strata and normalized overcormg depth
Fig 6 Dipole strengthJ as a function of overconngdepth h/a Dipole
h/a in thts figure is markedly &fferent from that in Fig 4 where
strength is normalized by the factor 8ha"<7o
0*] ~ <72 ~ <7(t
W O N G and W A L S H
of the major prlnctpal stress
12
~ - 0 45
%to
tan 20t =
,~=
o~
O6 04 -~ 0 2
~ oo -02
167
T E C T O N I C STRESS R E L I E F D U R I N G O V E R C O R I N G
0
I 1
I 2
I 3
I 4
I 5
Normal=zeal core depth ( h / o )
F~g 9 Stress a~(= - a , , ) at (0, 0, 0) normahzed w~th respect to % for the case where the tectomc stress state ~s pure shear (a~ = -~r., = 30) h/a ~s overconng depth normahzed w~th respect to core radms a Note the weak dependence on Po~sson's ratio, m contrast to F~gs 4 and 5
therefore can be considered to be representative of the observations m th~s area The measurements (Figs 10a and 1 la) were made at two sttes (Reed and Museum) m the Blue M o u n t a m Lake-Racquette Lake area where m a n y earthquake swarms have been detected since 1970 [12] The rock is gne~sslc, and mtcroscop~c observattons show that samples from Museum have more mlcrocracks and a higher degree of amsotropy than those from Reed The cores were 152 m m (6 m ) dla. The strata-gauge rosette was onented m the north, southeast and southwest directions [11] The stram measurements are expected to have an accuracy of + 10 x 10 - 6 [10], and the uncertainty m the measurement of overcorlng depth can be as much as 6 4 m m (1/4m.) IT Engelder, private communicaUon, 1981] We use our theorettcal results to estimate tectomc stress from the rosette stgnals m the following way Let the bearing w]th respect to north of the m a x i m u m pnnclpal tectomc stress aj be denoted by ~t and the rosette signals m the north, southeast and southwest &rectlons be denoted by es, eSE, eSW respectively M o h r ' s orcle shows that the rosette strams are related to the principal strata e~ and e,, by
eSE = ( ~ )
-- ( e ~ 2 e " ) s,n 2~t,
(7)
Equattons (6) and (7) are the basis of our technique for mterpretmg Engelder and Sbar's observations. First we plot (eEW+ eSE)/2 as a function of overconng depth h/a We then estimate Polsson's ratio by c o m p a r m g the ratto of the peak stress to the asymptotic value from thts plot wtth Ftg 7 Values of ao/E and Potsson's ratto are then adjusted to find the best fit The observattons and the theorettcal curves for the Reed and Museum sites are compared tn Ftgs 10b and 1 lb We see that the fit to the Reed data is acceptable for a range of Polsson's ratios from 0 to 0 1, and the fit to the Museum data is less satisfactory Note that the observed average stram at Museum reaches a m a x i m u m at a depth greater than that predtcted by theory. This sort of mtsfit cannot be removed by adjustmg tro/E and v We thmk that the discrepancy between theory and observatton at Museum is probably due to antsotropy of the formation there: we assume that rock is ~sotrop~c m our analys~s, whereas Engelder and Sbar reported that Museum was noticeably amsotroptc, much more so than Reed The values of tro/E and v that we find using the average stram data are gwen m Table 1. To find a0, we esUmate E using our value of Potsson's raUo v and laboratory measurement of P-wave velocity by Engelder and Sbar [11]. Density needed to reduce the velocity data is calculated from modal analys~s We show m Table 1, somc velocity Vr, our calculated value of E, and the mean tectomc stress a0. Velocity vaned somewhat w~th ortentation, and so the average values are entered in Table 1 The next step is to evaluate the dewatonc component % of the remote stress and the onentat~on ~t of the m a x i m u m pnnc~pal stress relatwe to north We calculate the dewatonc strata ( e ~ , - e,,)/2 usmg the two expressions m (6). The theoretical expressions m (4) are fitted to the data by adjusting the parameters % sm 20t and % cos 2~ The raUo of the two adjustable parameters is then calculated to find tan 20t The compartson of theory and observaUon for the deviatonc component at Reed and Museum ts shown m
Table 1 In the upper table, the p n n o p a l tectomc stresses trl and a 2 and the orientation ~t with respect to north estimated m th~s study are compared w~th the values estimated by Engelder and Sbar [I I] Other reformation revolved in this study is given m the lower part of the table Reed
Rearranging (5), we find that the mean strain ( e . + ev~)/2 and the devmtonc strain (~. - e,,)/2 can be expressed m terms of the rosette stratus by (e~ + e,,)/2 = (esw + esE)/2
al(MPa) a2(MPa)
(ex~ - e,,)/2 = (esw - esE)/2 sm 2~t
tr0(MPa) %(MPa)
= [er~- (ese + esw)/2]/cos 2~t.
esw - eSE 2eN -- (eSE + eSW)
(6)
Note that the two expresstons for the deviatoric component m (6) can be combined to g~ven the bearing
ao/E(lO-6) v E(GPa) Vp(km/sec)
This study 58 ~ 63 55
Museum
E & S
This study
E & S
89 ~ 89 57
162 ~ 21 5 3 1
144' 21 6 13 8
5 9 (this study) 0 4 (this study) 72 (this study) 0 10 (thts study) 82 (this study) 5 67 (E & S)
12 3 92 205 0 05 62 4 88
(this (this (this (this (this (E
study) study) study) study) study) & S)
168
W O N G and WALSH
TECTONIC STRESS RELIEF D U R I N G O V E R C O R I N G (o)
(o) Reed
T
100
[11]
• T " J. 8O
•
• D
Q
60
c
o
D
200 D
•
co 150
u~ 4 0 •
•
SW
100
o N
20
•
SE •
50 I
I
I
I
I
1
2
3
4
5
Core
SW
0 N •
depth ( m )
SE
I
I
J
I
[
1
2
3
4
5
Core
depth ()n)
(b)
120
~
leo •, 80 ::t
(b) : !
Reed [11]
30O
Museum [111
250
N 20O
60 •
•
~
U,01
w
O"0 12 3 MPo
÷
%
u T
[3 •
, o
Od
Museum [11]
•
2,50
•
i
0 •
300
13
~u :L
o
350
2O
0
50MPo
E 8 4 GI=o
01
5,9MPo
82GPo
I
I
1
I
I
2
3
4
5
6
0
2o
J
I
I
I
I
2
5
4
5
6
Core d e p t h [ )n )
(c) Reed
[113
Museum I"11]
r0 0 4 M P o
• EN-(~SWt'~SE)/2
U 58"
o ((SW-ESE)/2
120
TO - 9 2MPo
a ~ l " ~
so 1o w
i
c 5
I
1
(m)
(c) 30
61GPo
5O
I
depth
0 05
E )
1
Core
u
. •
40
0
/
e,
e (N-((SW* (SE)/2 I
_~
0
"Z~.
I
I
~
I
I
I
I
' o (,(SW-~SE)/2
(~ - 4 0
-20 -30 1
I
I
l
I
1
3
4
5
6
Core
depth (=n)
-120 0
[
i
I
[
I
I
t
1
2
3
4
5
6
7
Core
depth ( ) n )
Fig 10 (a) Stratus in the north, southeast and southwest dlrectmns as a funcUon of overconng depth from measurements at Reed [11] (b) Data points are average normal stratus (E.~+ ~,,)/2 = (esw + esE)/2 recorded by the strata rosette at Reed [11] The curve Is our theoretical relatmn (equatmn 3), where we find that the best fit ts obtained when Po~sson's ratio ts m the range 0--0 1, and the average remote stress a0 is approxtmately 8 3 MPa (c) Dewatonc stram at Reed Sohd mrcles represent strata (%w--%E)/2 = (e~,--~,,) (sin 2~t)2 calculated from rosette stratus [11] Open circles represent strata component (e. - ~,,) (cos 2=)/2 = e~ - (%E + gsw)/2 calculated from rosette stratus [il] Curves are from equauon (3) for r 0 = 0 4 MPa and a = 58°
Fig 11 (a) Stratus m the north, southeast and southwest d=rectmns as a function of overcormg depth from measurements at Museum [I I] (b) Data po,nts are average stra,ns ( e ~ - e , , ) (cos 2~)/2 = eN - ( e s e + %w)/2 recorded by the strata rosette at Museum [11] The curve Is out theoretical expressmn (3), evaluated for Polsson's ratm equal to 0 05 and average remote stress % = 12 3 M P a (c) Devmtonc strata at Museum Sohd orcles represent strata component eN - (ese + %w)/2 = (e~-e,,) (cos2~)/2 and open orcles represent strata component ( e s w - esE)/2 = (e,,- e,,) (sin 2c()/2 Curves are from equatmn (3) for r 0 = 9 2 M P a and ~ = 1 6 2
F i g s 10b a n d l i b T h e r e l a t i v e l y p o o r fit for the R e e d d a t a is d u e to the small c o m p o n e n t o f s h e a r at this site a n d the r e l a t i v e l y large u n c e r t a i n t y a r i s i n g f r o m c o m p u t i n g the difference b e t w e e n t w o n e a r l y e q u a l n u m b e r s V a l u e s f o r t h e s h e a r c o m p o n e n t % a n d o n e n t a t m n ~t a r e e n t e r e d in T a b l e 1 H a v m g c a l c u l a t e d a 0 ( w h i c h e q u a l s (a~ + tr2)/2 ) a n d z0 ( w h i c h e q u a l s ( t r j - trz)/2), we n o w h a v e the t w o c o m p o n e n t s a~ a n d tr 2 o f t e c t o m c stress a n d the o n e n t a t l o n tr~ r e l a t i v e to n o r t h O u r v a l u e s are c o m p a r e d In
T a b l e 1 w i t h t h o s e d e t e r m i n e d by E n g e i d e r a n d S b a r T h e a g r e e m e n t , m g e n e r a l , ts s a t i s f a c t o r y T h e a p p a r e n t d i s c r e p a n c y in c( for the R e e d d a t a o c c u r s b e c a u s e the s m a l l s h e a r c o m p o n e n t m the r e m o t e stress state m t r o d u c e s a v e r y large u n c e r t a i n t y m the o r i e n t a t i o n , ~(, f o r e x a m p l e , is i n d e t e r m m a t e w h e n aj = a2. W e h a v e n o e x p l a n a t i o n for the difference b e t w e e n the t w o v a l u e s o f a: at M u s e u m , o t h e r t h a n t h a t the d i s c r e p a n c y m a y be d u e to the m a r k e d a n l s o t r o p y at this site
WONG and WALSH
TECTONIC STRESS RELIEF DURING OVERCORING
169
DISCUSSION coring depth that could not be observed in practice The We present above an analys~s of the direct strain- curves display no distinguishing characteristics which gauge technique for measuring m S l t U stress. We s~mu- can be used to estimate Poisson's ratio when one considlated the operation by assuming that the stress rehef due ers the inherent uncertainty m the data Consequently, to overcormg ~s equivalent to imposing d~splacement the s~mplest approx~matlon for the d~pole distribution is dlscontmmues which are uniform with depth on the adequate here, where data and theory are compared only for the purpose of checking results. cylindrical surfaces of the core and surrounding rock The comparisons between field measurements at two We found that the strain change observed at the surface of the formation on the centre hne of the core can be s~tes In New York State and theoreucal behavlour that expressed m terms of the remote stress state, the elastic we show in Figs 10 and 11 are considered to be properties of the rock, and simple aigebrmc funcUons of satisfactory m a field experiment of th~s sort Better agreement could possibly be obtained if the effects of the overcoring depth (see equations 1-4) In the analysis, the cases of an equal blaxml remote amsotropy could be evaluated. Most rock is found to be stress state and a stress state of pure shear were treated elastically amsotrop~c to some degree, and, as we discuss, separately, the general case being a superposmon of the the Museum s~te was noticeably so We looked m a two Two approximations were considered m the analy- prehmmary way at the analys~s needed to extend our sis of the case of equal bmxml remote stresses. For a first work to amsotroplc material and dec~ded that the potenapproximation, we assumed that the d~pole strength tml benefits were not sufficient to justify the enormous (which ~s uniform on the core surface) does not vary w~th amount o f calculauon revolved Amsotropy remains a coring depth, the magmtude of the d~pole strength was problem of practical ~mportance, which must be adcalculated from the value at mfimte coring depth, where dressed if we are to improve the accuracy of our the solution ~s known For a better approximation, we esumates of tectonic stress adjusted the d~pole strength (which again ts uniform over As discussed m the text, Polsson's ratio for the the core surface) such that the stress change on the core formation can be estimated from the stram data m surface near the free end equals the remote stress at all a d d m o n to the tectomc stresses and orientation We find coring depths that the value of Po~sson's ratio at Reed IS between 0 and The strain at the strata gauge at the centre hne on the 0.1 and the value at Museum ~s 0 0 5 Independent free surface ~s somewhat overestimated at small coring observations of PoIsson's ratio at these s~tes are not depths by the analys~s using the second approximation. avadable, but our estimated values are m the range Consider the stress change on the core surface for this reported for gneisslc rocks [13] at atmospheric pressure case the stress change at the free surface equals the Note that the values of Polsson's ratio are lower than remote stress but, to maintain the c o n d m o n that the those for the constituent minerals or for these same displacement change over the core surface ~s uniform, rocks at depth because Polsson's ratio Is lower for the stress change increases w~th core depth, becommg matermls which have a higher density of open microcracks [6] very large where the core is attached for the formation The average change m stress, and hence the strain at the We conclude from our analys~s of the d~rect strata gauge, ~s therefore larger than the value when the stress gauge techmque that curves of stram as a function of change ts uniform over the core surface The accuracy ~s coring depth provide addmonal reformation which Is poorest at small coring depths, but th~s ~s the region pertinent to the interpretation of field data. These results where measurements are least accurate because strains suggest that other stress-rehef techmques, particularly are small and weathering of the formation affects rock the closely related "door-stopper" techmque, should properties The approximation ~s certainly adequate also be analyzed In our analysis, overcormg must be here, considering uncertainties m the data and the mmated on a previously uncored half-space, and so our overall accuracy of the stress estimate that ,s needed. results cannot be used for even an approximate evaluWe used only the first level of approximation m our ation of the accuracy of these other techmques. calculations for the case where the remote stress state .s Acknowledgements--A lecture by Terry Engelder at M I T on stress pure shear The analys~s reqmred when the second measurement was the initial spark for this analysts, and he provided approximation ~s employed ~s lengthy because data and comments throughout the course of the study Dunng the early stages of the analysts we were supported by funds from the Army s~mpllficat~ons arising from symmetry m the equal bmx- Reserach Office (Durham) under Contract DAAG29-79-C-0032 and ml remote stress case are not avadable Fortunately, a the U S Geological Survey under Contract 14-08-0001-19792 The lower level of accuracy ~s acceptable for th~s component National Science Foundation (Grant EAR-8018493) prowded support to fimsh the first version of the analysts and manuscript A rewewer of the remote stress state In the first place, m s t t u tensile pointed out what seemed to be an easdy-remedted error which led stresses are virtually never observed, and so the shear eventually to an extensive reanalysls of the problem and revision of the component IS always less than or, at most, equal to the manuscript The recent work by one of us (J B W ) was supported by the Army Research Office (Durham) under Contrast 00-142-5594, the average normal stress component Also, less information U S Geological Survey under Contract 14-03-0001-21803. and the can be obtained from measurements of strata as a National Soence Foundation (Grant EAR 8218769) and by the other function of coring depth for the remote shear stress author (T-F W ) by the National Science Foundauon (Grant EAR8218379) We thank Terry Engelder and the reviewer for their valuable component Note m Fig 8 that strain at the gauge contributions to this study increases monotonically w~th mcreasmg coring depth, except for a region for low Potsson's ratio and small Recewed 26 October 1981. rewsed 20 August 1984 RMMS
223--D
170
W O N G and WALSH
T E C T O N I C STRESS RELIEF D U R I N G O V E R C O R I N G
REFERENCES
fh 12n
1 McGarr A and Ga) N C State of stress in the earth's crust 4 Ret Earth Planet Sct 6, 405-536 (19781 Zoback M L and Zoback M State of stress in the coterminous Umted States J GeophLs Re~ 85. 6113-6155 (19801 3 de la Cruz R V and Raleigh C B Absolute stress measurements at the Rangely ant]chne Northwestern Colorado lnt J Rm~ ~te(h Mm S(I 9. 625-634 11972) 4 Swolfs H S Handm J and Pratt H R Field measurements of residual strain m granitic rock masses 4dtante~ m Ro(~ Me(hanu* Prm 3rd Congr ISR~.I. Vol II pp 563-568 (1974) 5 Plumb R Engelder T and Sbar M Near surface ln-~uu stress II a comparison with stress directions inferred from earthquakes joints and topography near Blue Mountain Lake New York J Geoph~ ~ Re~ 89, 9333-9349 (19841 6 Walsh J B The effect of cracks on Po~sson s ratio J Geophls Res 70. 5249-5958 (19651 7 Weaver J Three-dimensional crack analysis hit J Sohds Stru(tures 13, 321-330 (19771 8 Brady B H G and Bra} J W The boundary element method elastic analysis of tabular ore bod) extraction, assuming complete plane strata Int J Rm~ ~lech Mitt S~I & Geomech 4b~tr 15, 29-37 (1978) 9 Mmdhn R D Forces at a point In the interior ot a seml-mhmte sohd P h ~ u ~ 1. 195-212 (19361 10 Engelder J T and Sbar M k Evidence for uniform strain onentatlon In the Potsdam sandstone northern Nev. York from m sttu measurements J Geoph] ~ Res 81, 3013-3017 (1976) 11 Engeleder J T and Sbar M L Stram relaxanon measurements in the vicinity of New York state using surface over-coring techtuques Technical Rept for Ne~ York State Energy Reserch and Development Authont3 38 pp 11977) 12 Sbar M L Armbruster J and Aggarv~al "~ P The Adirondak New York earthquake swarm of 1971 and tectonic Imphcatlons Bull Setsm So~ Am 62, 1303-1317 (1972) 13 Clark S P Jr Handboo~ o! Phvsual Con~tant~ Geol Soc Am Mem 97 587 pp (1966) 14 Gradshteyn I S and Ryzh]k I M Table o/bltegral~ Serle~ and P~oduJs 1086 pp Academic Press New '~ork (1965)
i ) [(7~ 4- 0"~
Stre~
-°'
0
(71 = a~ = (7.
] -- -~,I~
2
-
' cos20 - o,
Ii
(A1)
where v is Potsson s ratio and subscripts 1 2 and 3 refer to the ~ } and _- dlrecnons respectlveb We are interested in the stress and strata at the centre of the free end of the core ne at the origin of the unpnmed co ordinate system We also need the expressmn for stresses at the rim of the core te at (L } z ) = (a 0 0) m order to constrain the dipole strength F v s t we derive a general expression for the stress at an arbitrary point on the free surface due to a set of &poles wnth relative magnitudes given by (AI) If such a stress field ((7,, (7 ( 7 , ) for umt &pole strength is known m the primed co-ordinate system then the stress m the fixed unprlmed co-ordinate system due to a continuous dtstnbunon [ of dipoles on the cylindrical core surface are
sin20
]
adOd~,
7
I a , 2 -"3 o, L sin 20 + (7~ t cos
20/adOd~ /
(A2)
Because ot the symmetry m the remote stress field the stress rehef due to overconng must satisfy the condmons o,, = a , and a,, = 0 The first condmon reqmres that /I p
,
L o ~ 2 0 + a , ~ sin 20 adOd~ = 0
I
and hence
Mmdhn [91 calculated the stress (~,]',.(71'11 lCr~,:~ cr~,'IL and ((7~,. cr~I ). due to single point forces located at P m the ~' ~ ' and --' dlrectmns respectively (Mmdhn [91) The stress due to the combmatmn of &poles which are needed here are obtained by dlfferentmtmg Mmdlln's results as follows
a,, = -
f\
]
~ I'l
3~
I-~
?l'
I -- ~
"- i - v
f(
g(
(A4)
Performing the above operation ~e arrive at the Iollowtng expression (l+v)(l-2~)
o,, +o,,
4roll - ~ ):
I
We calculate the strain measured during overcormg by superposmon of fundamental solunons for point-force &poles distributed over the cored surface A thorough mvestlgauon of such point-forces in an elasnc half-space was performed b~ Mmdhn [9] For convenience of d]scusslon we use two co-ordinate systems mterchangeably (Figs 2 and 3) The unpnmed system is oriented parallel to the principal stresses assuming one pnnctpal stress to be vemcal The primed system is ldenncal to that used by Mmdhn Consider a point P on the cored surface with co-ordinates gwen by (~ ~ :')=(O,O()or(x ] z)=(acos0, asln0 ~)whereatsthe radius of the core and ~ ts the depth below surface If the m s]tu stress field ~s such that the two horizontal principal stresses are equal (at = a z =(7.L then s~mmetr} constrains the displacement discontinuity distribution on the o~ercored surface to be axnsymmetrlc It can be shown [7 8] that the elastic held for a radml displacement dnscon tmmty Is equivalent to that due to an orthogonal set of three &poles such a set of dipoles at P is shown in Fig 2 for the (x t z') co-ordinate system Weaver [7] and Brady and Bray [8] show that the relative magmtudes of the three components are given by the relatmn I =/,=
- cos 20 + a,, sm 20 J adOd~
2,1-~) ~11 2 , ) , 6(l-v), R~ Rz R"
APPENDIX Remole
2~
'3 e'h ~2-
2
]
+
2
15, "
~ \ "1 R"
(A5)
where R ' = : -'+~ -'T( The dipole strength/L ts m general a function ol both thc o',erconng depth h and the position ( To stmphfy the calculat[ons we assume that the &pole strength Is constant o~er the surtace ol the core although it may var) with coring depth m other words, 1~ is a functmn o f h only By substituting (A5) into IA3) we find after considerable algebra, analytmal expressions for the stress at tv.o locations of interest The first location is at the centre ol free end of the core (x = a ~ ' = 0), for which we obtain a,,=-
/ d h ) l l + ~ ) ( i _21 ) I--" , ( " - _ ~ )i~ 6h 3h ~3 2a I1-~1: L ,/ - d + d'/]
(A6)
where d z = a: + ll: The second location ~s at the rim ol the tree end of the core (,,' = a ( l + c o s 0 L i ' = a ~,tn 0), for which we obtain a,
=/~!h) ~ ~ 2_,, !!_L :_2i, i l~ 4ha
(
1-
~ 12
~}
(A7)
where I(k, v) -
x 4a: + h:
4at--t--
(5 - 41 ) I +L4
2/i" 4a:+
1
hZJE(k
))
and ,~ = 2 a \ 4 a Z + h h~/,) and E~/,) are the complete elhptlcal integral of the first and second kind The dipole strength can no~ be determined by imposing the boundary condition that a , = -(7o 2 at the second location ao = ll(ll) (1 + I )(1 - 2~) i(/,. ~ } 2 4~a ( I - ~ )2
where the factor of 1 2 is needed to take into account the jump in the boundary condmon at the free surface of the core, and the negative sign indicates that stress rehef is opposite m direction to the remote stress Hence, dipole strength is given b~
W O N G and W A L S H
f,(h)
-
2na(l - v)"
a0
(A8)
(1 + v ) ( I - 2v)I(k, v)
G, = g
On substituting (A8) into (A6), we a m v e at the final expressions for the measured stresses and stratus rtao [ 2 ( 2 - v ) h
~'~:~":~L G , = e,, -
n(1 v)T-
6h 3
J
a'
I(k, °'°,~) [_2(2 V - )hd
171
T E C T O N I C STRESS R E L I E F D U R I N G O V E R C O R 1 N G
3h ~-] 6h'd,+ a-3-jh; z'.l
(A9)
sin 20
[°
"' 2 0"vr --O'~ L
-~ -
a,,=
+-~J"
f0f
-
2
fofo' [°+° I'T" E sin20
cos 20 + G, sin 20] adO&.
~ 2
o,~ - a , , cos 20 - G , s,n 2
a,, = ~
sin 20
T
do do
2ol adOdc, J
sin 20
~
+ o , , cos 2 0 ] adOdc H
R e m o t e Stress tr I ~ a 2
The problem can be solved by superposltlOn of the solutions to the following two problems (1) a, = a, = a o, where a 0 = (a I +a,_)/2, and (2) a, = - o , = r o, where z o = Itr I - a , ) / 2 The first problem was considered m the prevmus section, and so we must consider only the case of pure shear here At a point such as P (Fig 3), one would expect to have nonvanishing &splacement discontinuities Au t and Au, in both the radml and azimuthal dlrecuon Futhermore, the distribution of &scontlnUltles is no longer axlsymmetrlc, and we expect from M o h r circle construction that Au, and Au, vary as cos 20 and sin 20, respectively The Au, component contribution is evaluated following the procedure described in the previous section but taking into account the cos 20-dependence We find that the Au, vamshes when we integrate over 0 from 0 to 2n The elastic field for the azimuthal displacement discontinuity is equivalent to an orthogonal set of two &poles with m o m e n t [7], often referred to as a double couple The double couple is shown in Fig 3 with reference to the p n m e d co-ordinate system The magnitude of the two components are equal and Is denoted by g = ~ sin 20. where ~ is independent of 0 The intensity g, Is In general a functmn of both the total overconng depth h and the position c We assume, as In the previous case, that ~, does not vary with t Prehmlnary analysis showed that an enormous Increase in the length of the calculations would be reqmred if the dipole strength were allowed to vary with coring depth Therefore, we assume ~, to be constant, justifying the a p p r o x l m a u o n for the reasons referred to m the text Once we calculate the stress (a, t. a , , . a , , ) In the primed coordinate system due to a set of double couples of umt strength located at P, the stresses In the fixed, u n p n m e d co-ordinate system due to a dlstrlbutlon of double couples on the cylindrical core surface are
(AIO)
Because we are interested only In the stresses at the centre line of the core, symmetry requires that at,=
-o,, = no~ J]'a,, d~
(All)
The shear stress tr~l and a~,-'b, due to a single point force acting at P in the x ' and ) ' distinctions respectively are denved by Mindhn [9] in his equation (18) tr,, for double couples IS obtained by differentiating M m d h n ' s expression ,'~ = _ ( 2 o ' , ''l) I + 2a~2"~ _ (AI2) a~, \ 2~' 2 r ' J" or exphc~tly
~"
1 [2(1-2v) 3a 2 = - ~ ~ / T ~ R'
6(1-2v)] ~+-~,j
(AI3)
/
evaluated at the centre of the free end of the core SubsUtutlng (AI3) into (A11). and integrating, we a m v e at [(5-4v)h_ 6(I-2v)h ~1 (AI4) = - o , , = - y~ L ~ a+ h d'J The double couple strength ~ ts determined by requlnng a~t to approach the asymptotic value - r 0 as h becomes very large
..
= - ~a (1 + 2v)
(AI5)
Substituting (AI5) Into (AI4) gives a,, = - a , ,
z0 [ ( 5 - 4 v ) h 1 +2v d
6(l-2v)h d+h
h ~] ~q
(AI6)
The expressmn for the measured strain is therefore £vx =
--/;~1
r0(l__+v) r ( 5 - 4 v ) 1 1 6(l-2v)h E(I+2v)L d + d+h
h~] ~q (AI7)