In situ stress measurements in anisotropic rock

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 21, No. 6 pp. 327-338, 1984

Printed in Great Britain. All rights reserved

0148-9062;84 $3.00 + 0.00 Copyright ~. 1984 Pergamon Press Ltd

In Situ Stress Measurements

in Anisotropic Rock B. AMADEI* The purpose of this paper is to discuss how to account for rock anisotropy when measuring in situ stresses by overcoring with instrumented devices such as the CSIRO hollow inclusion cell or the U.S. Bureau of Mines (USBM) gauge. The rock is described as an isotropic, a transversely isotropic or an orthotropic material. The computation of in situ rock stresses with the two devices mentioned above depends on the anisotropic rock character, the degree and type of rock anisotropy, the orientation of the rock anisotropy with respect to the hole 01 which the devices are located, and, for the CSIRO cell only, dimensionless parameters describing the relative deformability of the rock with respect to the deformability of the cell material. Using these parameters, the paper addresses the following problems: the effect of the degree and type of rock anisotropy on in situ stress determination; the error involved in neglecting rock anisotropy by assuming isotropy; the error involved in neglecting the geometry and properties of the CSIRO cell by assuming that its strain gauges are positioned directly on the walls of pilot holes, and how this error is affected by the rock anisotropy.

INTRODUCTION Any undisturbed rock mass is subject to natural stresses including gravitational stresses due to the mass of the overburden and possibly current tectonic stresses due to the straining of the earth's crust and remanent stresses due to past tectonism. Knowledge of the in situ stress field must be integrated into any rock engineering design, along with other general rock mass characteristics such as deformability, strength, permeability and timedependent behaviour. For example, the choice of optimal orientation, location, dimensions and shape of tunnels, mine openings and deep underground caverns or complex underground works will be controlled by the orientation and magnitude of the in situ stress field if it is necessary to minimize stress concentration and stability-related problems. Several overcoring techniques have been proposed in the rock mechanics literature to measure the state of stress in situ. A review of these techniques, as well as a discussion on their respective advantages and disadvantages, can be found in Goodman [l]. When dealing with overcoring techniques, the in situ state of stress is not determined directly, but instead strains or displacements are measured by disturbing the state of stress in the rock and these are related to the in situ stress components through closed-form solutions. In these solutions, the rock is usually assumed to be modelled as homogeneous, continuous, linearly elastic and isotropic medium.

*Department o f Civil Engineering, Boulder, CO 80309, U.S.A.

University of Colorado,

Additional assumptions are often postulated concerning (i) the orientation of the principal stress field with respect to the directions of the borehole in which the measurements take place, (ii) the mode of rock deformation (plane strain, plane stress) when overcoring takes place and (iii) the nature of the stresses that are computed (overburden stress, tectonic stresses, residual stresses). The final assumptions may become so restrictive that they may lead to erroneous in situ stress determinations. No solutions exist to date in rock mechanics that account for combined "non-idealistic" rock properties such as heterogeneity, anisotropy, discontinuous characteristics and non-linear elastic behaviour. However, solutions have been proposed in the literature to account for rock anisotropy when measuring in situ stresses in metamorphic rocks such as slates, gneisses, phyllites or schists or sedimentary rocks such as sandstones, limestones or shales. These solutions show different degrees of assumption and sophistication [2-9]. The purpose of this paper is to discuss how to account for rock anisotropy when measuring in situ stresses with instrumented devices such as the CSIRO hollow inclusion cell [10] or the USBM deformation gauge [I 1]. The paper begins with a review of the conclusions that have been drawn in the general theory for overcoring in anisotropic rock masses proposed by Amadei [9]. It is shown how the analysis for the computation of in situ rock stresses for overcoring measurements with the two instrumented devices mentioned above can be expressed in terms of dimensionless parameters describing the deformability of the rock and, for the CSIRO cell only,

328

AMADEI:

STRESS MEASUREMENTS IN ANISOTROPIC ROCK

dimensionless parameters describing the relative deformability of the rock with respect to the deformability of the cell material. This provides the basis for (1) a better understanding of how in situ rock stress determination in anisotropic ground is affected by the degree of rock anisotropy, and, (2) assessing the error involved in neglecting rock anisotropy by assuming isotropy. OVERCORING IN ANISOTROPIC ROCK MASSES

Overcoring as a relief technique

General formulae.[or overcoring in anisotropic rocks

Overcoring techniques can be classified as relief techniques, i.e. methods of procedures that wholly or partially isolate a rock specimen from the stress field in the surrounding rock. Strain or displacement measurements on the specimen thus isolated are recorded in the vicinity of the point at which the in situ state of stress has to be determined. This requires the stress field to be homogeneous throughout the region of the rock mass of interest. Figure t illustrates the three basic steps commonly involved when measuring in situ stresses by overcoring with instrumented devices such as the CSIRO hollow inclusion or the USBM gauge. First, a relatively large diameter hole, usually 150 mm dia, is drilled in the volume of rock where stresses have to be determined. The hole is drilled from the ground surface, or more often from an existing underground excavation, to a distance far enough so that the effect of any free surface on the stress measurements can be neglected. Then, a small pilot hole, that is usually 38 mm (EX size) in diameter, is drilled at the end of the previous hole. The instrumented device is inserted into the hole. The pilot hole must be long enough so the effect of its own ends and the disturbance in stress caused by the larger hole to the in situ stress measurements can be neglected. Large d~ometer hole

instrumented \'~/ I

I~"y/It

fr

"

device

",'f,'-

"

I

Pilo~

r:

i ss

Finally, coring of the large diameter hole is resumed and the resulting changes in strain or displacement arc recorded. Up to nine changes in strain are recorded with the CSIRO cell, and one borehole can be used to determine the complete state of stress. When using the USBM gauge, changes in up to three diameters of the pilot hole can be recorded per borehole and two or three non-parallel boreholes are needed to determine the complete state of stress.

hole

;F

I

Computation of in situ rock stresses from overcoring measurements requires that an analytical solution exists relating the strains or displacements measured during overcoring and the components of the in situ stress field. For both isotropic and anisotropic rocks, this can be done by using the closed-form solutions proposed by Amadei [9]. As shown by Amadei [12], these solutions can also be expressed in terms of dimensionless parameters describing the deformability of the rock or, for the CSIRO cell only, dimensionless parameters describing the relative deformability of the rock with respect to that of the cell material. Consider the equilibrium of the region of a rock mass that can be modelled as a linearly elastic, anisotropic, continuous and homogeneous medium. The medium is bounded internally by a cylindrical surface of circular cross-section that represents a borehole. The hole contains a hollow inclusion with outer and inner radii a and b respectively. The inclusion is assumed to be linearly elastic, isotropic, continuous and perfectly bonded to the anisotropic medium. Furthermore, both the hole and the inclusion are assumed to be infinitely long. Consider the geometry of Fig. 2 and let x, .v, z be a Cartesian co-ordinate system with the z-axis defining the longitudinal axis of the hole. The orientation of the hole and therefore the x-, y-, z-axes are defined with respect to a fixed arbitrary global co-ordinate system X, Y, Z. Similarly, the anisotropic medium has planes and/or axes of symmetry with respect to directions independent of the x-, y-, z-directions. Thus. let x', y', z' be a Cartesian co-ordinate system attached to the anisotropy. From a practical point of view, the x', y', z' co-ordinate system is attached to apparent planes of rectilinear anisotropy or symmetry in the anisotropic medium. For rocks, these planes are, for instance, schistosity, cleavage or bedding planes or apparent directions of rock layering. In any case, x ' is taken normal to the planes, whereas the y'and z'-axes are contained with the planes. The orientation of the rock anisotropy, and therefore that of the x'-, y'-, z'-axes with respect to the X-, Y-, Z-axes, is assumed to be known and to be defined by the two angles/7 and qs shown in Fig. 3. A general form of the constitutive relation for the anisotropic medium in the x, .v, z co-ordinate system can be written as follows: (~)<,. = (A) (o'L,_.,

Fill. 1. Steps commonly involved m overcorin t lechniques wiih inslrumenled devices such as the C S I R O ¢¢11 or lhe U S B M gauge.

(I)

where (c) ..... and (~),,= are rcspectively (6 × 1) column matrix representations of the strain and stress tensors in

AMADEI: STRESS MEASUREMENTS 1N ANISOTROPIC ROCK z

Y

Y

329

its deformability can be described by the following five independent elastic propertiest

x

E~; E2 = E3; v21 = v3,; v32;Gl: = Gl3,

(2)

E., E2, E3 are Young's moduli in the x'-,y'-.:'-directions respectively, Gt2, Gt3 are shear moduli in the x'v'-, x'z'-planes respectively and v,j where

o

x I/

//1

determines the ratio of strain in the j-direction to the strain in the i direction due to a stress acting in the /-direction. v,j and vj, are such that v,/E, = v,,/Er In the x, y, : co-ordinate system, the same medium has, at most, one plane of symmetry normal to the :-axis and its deformability is now described by 13 elastic constants or compliances that depend on the five elastic properties of equation (2) and the value of the angle ~ defining the inclination of the planes of transverse isotropy with respect to horizontal. These 13 coefficients reduce to 5 when ff is equal to if: or 90 :. Similarly, in Fig. 4b, the medium is again transversely isotropic in the x ' , y ' , : ' co-ordinate system but its deformability in the x. v. : co-ordinate system is now described by 21 elastic constants or compliances since the planes of transverse isotropy are inclined with respect to the global coordinate system and the hole. These 21 terms depend on ~z,o Tzy,o the values of the angles fl and tk defining the orientation Fig. 2. Geometry of the problem---elasticequilibrium of an infinite of the planes of transverse isotropy with respect to the anisotropic medium bounded internally by a isotropic hollow hole and on the five elastic properties defined in equation inclusion. (2). The same remarks would also apply if the medium in Fig. 4 were orthotropic with the following nine elastic the x, y, z co-ordinate system and (A) is a (6 x 6) constants in the x', y', : ' co-ordinate system. compliance matrix whose components a,j (i, j = 1-6) can Et, E:, E~, GI2, G,.~, G2~, v21, v3,, v32. (3) be calculated from those of the compliance matrix in the x', y', z' co-ordinate system. In general, matrix (A) has The anisotropic medium of Fig. 2 is loaded at infinity 21 distinct components, but this number is further by a three dimensional stress field whose matrix reprereduced if the anisotropic medium possesses any sym- sentation in the x, y, : co-ordinate system, (o 0), is such metry in its inherent structure that also appears in its that mode of elastic deformation in the x, 3', - co-ordinate (a0)' = (a~.0 a,.o o:.0 %--.0r~:.(, r~,.0). (4) system. The number of distinct components ¢, is equal to 13 if the medium presents a plane of elastic symmetry General closed-form solutions between the components perpendicular to one of three x-, y-, :-axes, 9 if the of matrix (a0) and the components of stress, strain and medium is orthotropic (i.e. possessing three planes of displacement at any point (r/a, O) in the inclusion have symmetry, each one being perpendicular to a co-ordinate been derived by Amadei [9]. New expressions for these axis), 5 if the medium is transversely isotropic (i.e. solutions have recently been proposed by Amadei [12] isotropic within a plane perpendicular to one of the three and are summarized below. Assume that the inclusion co-ordinate axes), and 2 if the medium is isotropic. It is has a Young's modulus E and a Poisson's ratio v and noteworthy that except for the isotropic case, these that the anisotropic medium is orthotropic with the nine numbers apply only when the co-ordinate system x,),, = elastic constants defined in equation (3). The relation is attached to the material symmetry directions associ- between the strain components in the cylindrical r, O, z ated with the x',y',,-'-axes. In any other co-ordinate system, the components of matrix (A) will depend on the 1Y compliances or elastic constants defined in the x', y', : ' p lone cf rectH,neor \k l~ ~ \, '\ / co-ordinate system and on the orientation of x, y, : with respect to x', y'. z'. As an illustrative example, consider ~--L-..\ \ \ \ ~k \ .< ". Fig. 4.* In Fig. 4a the medium is transversely isotropic in the plane y ' z' of the x'. y', : ' co-ordinate system and 11

* In this example, the x-, y-, z-axes are taken parallel to the global axes X, Y, Z for the sake of clarity. t"The shear modulus G,3 is not independent and can be expressed in terms of v~zand E,.

Z~

/\\\\/,\

~

,%.~ \\

'~

HOrLzontol in

"XO~plone

Fig. 3. Orientation of the co-ordinate system x ' , y ' , :" with respect to

the global one X, Y, Z. Definition of angles fl and ~.

330

AMADEI:

T

J

STRESS M E A S U R E M E N T S IN ANISOTROPIC ROCK

x

Z

Yl A

•

/

(b) Fig. 4. Influence of orientation of anisotropy on the deformability properties in the x,.v, z co-ordinate system. (a) Inclined transverse isotropy parallel to z-axis. (b) Inclined transverse isotropy non-parallel to ,--axis.

co-ordinate system at any point in the inclusion and the components of the applied stress field can be expressed as follows.

E E, E I El

~o

"]o:

1 = = (Q,)(ao). nl

(5)

I rz [rO

Similarly, the three components of displacement in the

r,O,z co-ordinate system are such that Ur m

a L.0 a

orthotropic planes of symmetry with respect to the inclusion, the co-ordinates (r/a, O) of the point of interest and the following nine dimensionless parameters*

1 t:,

= - j (Q.)(ao).

El

E,

E,' E2" E3" G,3" G,:" G23" v2'' v31' v~2"

(7)

These parameters describe the relative deformability of the anisotropic medium with respect to the deformability of the inclusion. If the anisotropic medium is transversely isotropic with the five elastic properties of equation (2), then the nine parameters in equation (7) reduce to the following ones E El El

(6)

ll'

a

(Q,) and (Qu) are respectively (6 x 6) and (3 x 6) matrices whose components depend on the Poisson's ratio and geometry of the hollow inclusion, the orientation of the * Due to the complexity of the closed-form solutions, the general expression for the components of matrices (Q<) and (Q,) is not presented in this paper. This can be found in Chap. 5 and the corresponding appendices of Ref. [9].

El" E2" Gl2' v21, v32.

(8)

Finally, if the medium is isotropic with two elastic properties Ej = E2 and v2t = v32, then these five parameters further reduce to two, e.g. E

E--~l'v'l"

(9)

Recalling the general overcoring procedure described in Fig. 1, the process of overcoring can be seen as cancelling the components of initial stress acting across

AMADEI:

331

STRESS MEASUREMENTS 1N ANISOTROPIC ROCK

a cylindrical surface in the rock. Theoretically, the size of the overcoring diameter ought to influence the distribution of stresses and strains within the overcored rock sample and what is actually measured in the instrumented device. However, the finiteness of the overcoring diameter can be neglected and set to infinity for the following conditions [9,12]: (i) the instrumented device in contact with the rock does not interfere with the deformation of the rock during overcoring. The USBM gauge falls in this category. The overcored sample is completely free of stresses and strains; (ii) the instrumented device does interfere with the deformation of the rock but it can be described as a soft, thin-walled, hollow inclusion. The CSIRO hollow inclusion cell falls in this category as long as the rock in contact with the cell is not too soft. For these conditions, the overcored sample is free of stresses and strains except in a region near the contact between the rock and the inclusion. Let the in situ stress field (tr0) be defined by equation (4) in an x, y, z co-ordinate system attached to the hole where overcoring measurements take place. When either one of the above conditions (i) or (ii) is satisfied, the process of overcoring can be seen as applying a 3-D stress field equal to -(tr0) at infinity. Therefore, if the instrumented device that is used in the overcoring techniques can always be described as a hollow inclusion, equations (5) and (6) can be used to relate respectively changes in strain or displacement measured during overcoring with the components of the 3-D m situ stress field. In these equations, (~0) must be replaced by -(a0). Since overcoring measurements may take place in one or several boreholes, it is more appropriate to rewrite equations (5) and (6) in terms of the m situ stress field (tro)xvz expressed in the global co-ordinate system X, Y, Z shown in Fig. 2. (ao)xrz is linearly related to (%) as follows:

(ao) = (To)(ao)xrz,

and, the angle g,~ between the tangential axis, 0, passing through the centre of the gauge and its longitudinal axis 1, (Fig. 5b). In the current model of the cell, the three rosettes are spaced at 120° along the circumference. The arrangement of strain gauges gives three circumferential or tangential strain measurements (~,, = 0:), two axial strain measurements (~,,=90°), and four additional strain measurements, three with ~k,= 45 ~ and one with ¢i = 135 °. Assuming perfect bonding between the CSIRO cell and the rock, equations (5) and (10) can be combined and used to relate the strains measured in the cell and the components of the 3-D in situ stress field. For a gauge, i, whose centre is located at an angle 0, from the x-axis and whose longitudinal axis is inclined at an angle ~, with respect to the 0-axis, we have 1

-~t, =-~l(cos20,sin2d/,sin~,cos¢,)(Ts)(¢ro)xrz,

where E~, is the strain measured in gauge, i, during overcoring. (Ts) is a (3 x 6) matrix whose components depend on the components of (T,) defined in equation (10) and on the components of the second, third and fourth lines of (Q,) defined in equation (5). They are calculated at the point r/a = 17.5/19, 0,. Equation (12) can be written for each one of the nine strain gauges (i = 1-9). This will lead to a system of nine equations and six unknowns which can be solved for the least square estimates of the six components of the in situ stress field as well as their respective domains of variation for different desired degrees of confidence by using a Student's t-distribution [13]. In the previous analysis, we assumed that the strains measured in the CSIRO cell correspond to the strains induced by overcoring at the centre of the nine strain gauges. In other words, the length of the strain gauges was neglected. As shown by Amadei and Walton [14], RosetTe C

(!0) Rose))e A

where (%) is a co-ordinate transformation matrix for stress and (tro)xrz is such that (~r0)~'rz = (ax.0 ~r.0 az0 ~rz.o ~xz.o ~xr.o).

(12)

y~

to)

(I 1)

Analysis of in situ stress measurements for the CSIRO cell The CSIRO cell can be described as a thin-walled, epoxy, hollow inclusion in which strain gauges are embedded, Its outer and inner radii are 19mm and 16ram respectively and the strain gauges are located 1.5mm away from the borehole wall. In the current model of the cell, three triple-gauge rosettes enable l measurement of strains during overcoring. Let A, B and C be the three rosettes. Figure 5a shows their geometry (b) z and orientation with respect to the x , y , z co-ordinate system attached to the pilot hole in which the cell is installed. The position of each one of the nine strain Fig. 5. Geometry of the CSIRO cell. (a) Orientation of the nine strain gauges, i, (i = 1-9) can be defined by two angles: the gauges in the cell. (b) Orientation angles 0,, ~, for one of the nine strain angle 0~ between the x-axis and the centre of the gauge, gauges, -

332

AMADEI:

STRESS MEASUREMENTS IN ANISOTROPIC ROCK

the length of the strain gauges (10 mm) can be taken into account by integrating the right hand side of equation (12) over each gauge length and by replacing the left hand side of this equation by the measured strain. Then. the analysis is identical to the one described for point strains. In general, if more than one borehole is used for the determination of the in situ state of stress at a point. equation (12) or its equivalent for average strains can be derived for each gauge in each borehole cell and combined into a more general system of 9 N equations and six unknowns with N being the number of boreholes. This system can then be solved for the least square estimates of the in situ stress field components.

where t, (i = 1-6) depend on the components ot matrix (T,) defined in equation (10) and on the components of the first line of matrix (Q,) defined in equation 161. It is noteworthy that AD 2 a reduces to AD D when there is no inclusion in the pilot hole [a = h~ and changes in diameter are measured directly on the walls or" the pilot hole. According to equation (13). any change in diameter recorded during overcoring is a linear function of the six components of the in situ stress field. Therefore. determination of these components requires that we set up a system of six independent equations from the results o1" six measurements. A general expression for that system can be written as follows

Analysis o( in situ stress measurements.[or the U S B M gauge

(AD 1 = ~ (T) (al,)~w.

As shown by Amadei [9], equation (6) can be used for the analysis of in situ stress measurements obtained by overcoring techniques for which changes in diameter are measured in the pilot hole. Measurements of change in diameter can take place either on the walls of a hollow inclusion perfectly bonded to the surface of a pilot hole or directly on the walls of the pilot hole. The second procedure can be conducted with the USBM six arm deformation gauge which can monitor pilot hole diameter changes along three diameters simultaneously. The first procedure has never been reported in the literature and could be conducted by inserting the USBM gauge into a hollow inclusion (sleeve) that has previously been bonded on the pilot hole walls. The advantages of this procedure with respect to the classical method are that the diametral measurements are less affected by rock heterogeneities, discontinuities, water and pilot hole irregularities or weathering. Let D be the inner diameter of the hollow inclusion, AD be the change in length of that inner diameter measured at an angle 0 from the x-axis and, 2a be the outer diameter of the inclusion as shown in Fig. 6. During overcoring, AD is related to the six components of the in situ stress field in the X, Y, Z coordinate system as follows

where (AD) is a ( 6 × 1) matrix of measurement of changes in diameter, AD,2a, and (T) is a (6 x 6) matrix with components equal to the coefficients t, (i = I-6) of equation (13) calculated for each measurement. Since we have at most three independent measurements of changes in diameter per pilot hole [9], we need at least two non-parallel boreholes to solve for matrix (¢~0)xrz in equation (14). This applies as long as matrix (T) is non-singular. It was found by Amadei [15] that this singularity depends on three parameters: the angle between the two boreholes, the anisotropic character of the rock and the orientation of the two boreholes with respect to the planes and axes of symmetry of the rock. In any case for which (T) is singular, a third borehole is always needed. Whenever three or more boreholes are used for the determination of the in situ state of stress at a point, equation (14) becomes a system of more than six equations. The number of equations depends on the number of boreholes and diametral measurements conducted in each borehole. The system can then be solved for the least square estimates of the component of matrix (a,,).~.~.t.

AD

1

2a

El

(h O'x.0+ t2 o'r.0 + t30"z.o+ t4 Zrz.o + ts Xxz.o + t6 ~xr.o),

(13)

tY MeosuremenT of honge in d l o m e t e r

Fig. 6. Measurement of change in diameter using the USBM gauge.

I

(14)

Measurement of the elastic properties of anisotropic rock

The analysis of in situ stress measurements with the CSIRO cell or the USBM gauge in anisotropic rock requires a knowledge of rock deformability. This can be obtained by testing rock cores that have been sampled in the field at places located remotely from the in situ stress measurement site. The specimens are then tested in uniaxial or triaxial compression. However, this method may lead to additional uncertainties about the in situ stress determination due to possible rock material differences from one point to another in the rock mass. Another method, suggested by Amadei and Walton [14] for the CSIRO cell only, is to conduct biaxial tests directly on the overcored sample with the cell inside it. By applying a uniform pressure on the outer surface of the overcored sample, strains are recorded in the cell and are then interpreted in terms of the anisotropic elastic properties of the rock comprising the overcored sample through a system of non-linear equations. Using that

AMADEI: STRESS MEASUREMENTS IN ANISOTROPIC ROCK method which is an extension of the one used for isotropic rock [16], the deformability of the rock is measured directly on the sample used for the in situ stress determination.

INFLUENCE OF ROCK ANISOTROPY ON THE

ANALYSIS OF IN SITU STRESS MEASUREMENTS MADE WITH THE CSIRO CELL The strains measured in the CSIRO cell and the changes in diameter measured with the USBM gauge during overcoring can be analyzed and interpreted in terms of in situ stresses by using program NMP5 developed at the CSIRO division of geomechanics [I 7]. In this program, the rock is modelled as an isotropic material so rock anisotropy cannot be accounted for in the computation of in situ rock stresses. However, general rock anisotropy can be handled by the program ANISS recently developed by Amadei and Walton [14] that makes use of the general theory developed by Amadei [9]. Equations (12) and (13) and the dependency of the components of (Ts) and (Q~) on the value of the dimensionless parameters of equations (7), (8) or (9) suggest that the computation of in situ rock stresses when using a CSIRO cell or a USBM gage located on the inner surface of a hollow inclusion depends on (i) the anisotropic rock character, (ii) the degree and type of rock anisotropy, (iii) the orientation of the anisotropy with respect to the pilot holes and. (iv) the relative deformability of the rock with respect to the deformability of the cell or the inclusion. It is noteworthy that if the USBM gauge measures changes in diameter directly on the walls of the pilot hole, the parameter E/E~ vanishes in equations (7)-(9) and the in situ stress components calculated with equation (14) become proportional to El. Only terms describing the deformability o f the rock remain in equations (7) and (9). The same would apply if the strain gages in the CSIRO cell were positioned along the wall of the pilot hole and the cell geometry as well as Young's modulus of the cell material were to vanish. The analysis of stress measurements with such an instrumented device would then become identical to the one for the CSIR triaxial strain cell proposed by Leeman and Hayes [18]. In order to understand how in situ rock stress determination in anisotropic ground is influenced by the degree of rock anisotropy and to assess the error involved in neglecting rock anisotropy by assuming isotropy, a numerical example is now presented with the geometry o f Fig. 2 and for the CSIRO cell only. A CSIRO cell with Young's modulus E = 3500 MPa, Poisson's ratio v = 0 . 4 and ratio outer/inner radii a/b = 19/16 is located in a pilot note whose local co-ordinate system x, y, z is taken parallel to a fixed global co-ordinate system 2(, Y , Z . The X- and ( - Z ) - d i r e c t i o n s are taken parallel to the magnetic east and north directions respectively. The strains measured during overcoring and the corresponding strain gauge positions are listed in Table 1. The strain measurements

333

T~b.F~i~Strain gauge orientation and strain gauge measurement Angle Angle Strain Strain O,b ~O,b Strain rosette gauge () () measurements" A0 49.6 270.0 - 169 x 10-~ A A~ 26.7 180.0 -233 x 10-6 A4~ 26,7 225.0 -5 x 10-~ B4s 247.6 45.0 -291 x 10-~ B BL, 247.6 315.0 -386 x 10-~ B,~ 264.0 0.0 -521 x 10-" C0 167.4 270.0 - 153 x 10-~ C C~0 144.5 180.0 -248 x 10-~ C4s 144.5 225.0 -593 x 10-~ ~Tensile strains are taken as negative values. bSee Fig. 5. correspond to test A/7525 from Walton and Woromicki

[20]. For the given set of strain measurements, the least square estimates of the principal components of the in situ stress field try, tr,,a3 and their orientation with respect to X, Y, Z have been calculated for several rock properties: (1) The rock is transversely isotropic with the geometry of Fig. 4 and the elastic properties El, E,., Gt2, v21, v3., of equation (2). The orientation of the planes of transverse isotropy with respect to the cell or equivalently the x,)', : co-ordinate system is assumed to be known and defined by the two angles fl and ~ of Fig. 3. These angles can be interpreted respectively as strike and dip angles for the planes of transverse isotropy. Both angles may vary between 0 ° and 90L The domains of variation for the five dimensionless parameters of equation (8) are such that (a) E/E~ is equal to 0,4, 0.1 and 0.05. These values correspond respectively to E l = 8 7 5 0 M P a (low modulus rock), Et = 35,000MPa (high modulus rock) and Ej = 70,000MPa (very high modulus rock). (b) v32 and v2~ are equal to 0.25 and 0.27 respectively, (c) the domain of variation of E~/E,_ depends on the values of v21 and v3,. According to Picketing [19], the following condition must be satisfied for the strain energy to be positive E, LI

(I -

- 2v , > o.

(15)

For the values of v2~ and v32 considered above, this is satisfied if El~E2 is less than 5.14. Thus, the following three values of E~/E2 are considered: 2, 1 and 0.5, (d) E~/GI2 is equal to 2.19, 4.38 and 8.75. (2) The rock is isotropic. Its Poisson's ratio is taken as 0.25 and one of the three values of E~ considered above is taken as the rock Young's modulus. Thus, E~/E2 and Ei/Gu are equal to I and 2.5 respectively. The magnitude and orientation of the principal components of the in situ stress field computed with program ANISS by assuming the rock to be isotropic and point strain measurements are listed in Table 2 for each value o f E/Ej considered in this parametric study. The domains of variation of at, a2 and ch are also given for a

334

AMADEI:

STRESS M E A S U R E M E N T S IN A N I S O T R O P I C ROCK

Table 2. In situ stress determination, lsotropic solution. Point strain measurements O"I

O',

(MPa) Beannga

E .E I

Elev2

0" 3

( M l ~ a ~ Bearing~

Elev."

(MPa) Bearing"

Elev."

0.4

LS U95°,, L95 ~,,

3.3 3.4 3.2

29.5 27.9 30.6

24.7 23.9 25.4

2.0 2.2 19

106.7 10~.5 105.2

- 25.7 - 20.4 -2tL4

0.4 0.8 0.03

157.0 162.5 1542

52.~) 5"7 7 4q.3

0.1

12.7 LS U95'~o 13.0 L95% [2.4

27.2 25.1 28.7

25.2 24.1 26.1

7.4 7.g 7.1

104.5 105.9 103.1

-25.1 - 19.6 -28.8

1.6 2.9 0.1

155.9 161.0 153.3

53.1 58.1 4q.3

25.1 LS 0.05 U95°o 25.7 L95°o 24.6

26.8 24.7 28.4

25.2 24.0 26.1

14.6 15.4 13.9

104.1 105.6 102.8

-24.9 - 19.5 -28.8

3.2 5.9 0.3

1557 160.7 153.2

53.1 58.1 49.2

'Defined with respect to magnetic north. bDefined with respect to XOZ plane. LS: least square solution. U95°o: upper 95°o confidence limit. L95%: lower 95'!. confidence limit.

95~o degree of confidence. It appears that the magnitude of the principal stress field components a~, a,, a3 are almost proportional to the rock modulus E t and that their orientation is only slightly affected by the value of that modulus or the value of the ratio E/E~. It has been found that, if the least square solutions for the principal stress components obtained by using point strains are compared to those obtained by accounting for the strain gauge length, then the error involved in neglecting the finite character of the strain gauges does not exceed 5°o. This has been observed for the isotropic solution and the anisotropic one described below. When the rock is modelled as a transversely isotropic material and the degree of rock anisotropy is given, the in situ stress determination depends on the orientation of the planes of transverse isotropy with respect to the CSIRO cell. This is shown in Fig. 7 in which the magnitude and orientation of the principal stress field components aj, a2, a3 have been determined for different values of the strike and dip angles fl, ff when E/E~ = 0.1. El~E2 = 0.5, Ei/Gl2 = 4.38, v21 = 0.27 and v32 = 0.25. Similar variations have also been obtained when Et/E 2 is equal to 1 and 2. Table 3 lists for each value of Et,'~ considered in this analysis and for ~Oequal to 0 °, 30 ° and 90 c, the values of the ratio a~;/a~,,, a2;/a2,,, a3Ja3,,. In these ratios, a~,, a2, and a3, are the principal stress field components calculated with the isotropic solution. Their magnitude is given in Table 2. at,., o',~ and a3,, are the corresponding principal stress field components calculated with the anisotropic solution whose magnitude departs the most from the magnitude of a,, a2, and a3, respectively when/~ varies between 0 ° and 90 °. The three ratios introduced above can be used to calculate for each principal stress field component the maximum relative error involved in neglecting anisotropy by assuming isotropy. Table 3 gives also, for each value of q~, the maximum angle between the direction of each principal stress field component obtained with the isotropic solution (Table 2) and the corresponding direction obtained with the anisotropic solution when fl varies between i f and 90 ~'. Neglecting rock anisotropy can induce large errors as far as the magnitude and orientation of the principal m situ stress field are concerned, especially when E~/F_,,is equal to 0.5 or 2. The errors are less when

E~..'E2 is equal to 1 because the rock is closer to an isotropic material for which Ej/E, = 1 and E~/G~, = 2.5. For any fixed orientation of the planes of transverse isotropy with respect to the CSIRO cell, the in situ stress determination depends on the degree of rock anisotropy and on the relative deformability of the rock with respect to that of the cell material. In order to illustrate this statement, assume that the planes of transverse isotropy are striking in the north-south direction (fl = 0 ~) and dip at 3 0 to the east (~0 = 30:). Figure 8 shows the variation of the magnitude of the principal stress field components a~, a 2 and 03 with Ej/E2 for the three values of E~/G~2 considered in this study. The ratio E/E, is fixed and is equal to 0.1. For any fixed value of E,/G~2, a,, a 2 and 03 decrease as EdE2 increases. This variation can also be seen as a decrease of o,, a2, a3 as E/E, increases since E/E2=(E/E,)(E,/E2) and the ratio E/E, is constant. The figure also shows that for any given value of EJE2, aj and a, increase as E~/G,2 decreases. This variation can also be seen as an increase ofa~ and a: as E/Gt, decreases since E/Gt2 = (E/E1)(E1/G~,). However, this last trend is not always true for a 3. The errors associated with neglecting rock anisotropy by assuming isotropy are summarized in Table 4 for the three values of E,/E2 and E,/G~2 considered in this analysis. As far as the magnitude of the principal stress field components is concerned, the smallest errors take place for EdE2 = 1 and Ei/Gl2=2.19 (E/E2=O.I; E/Gt2=0.22) whereas the largest ones are for E,/E2=2 and E,/G12=8.75 (E/E2 = 0.2; E/GI2 = 0.87). Also shown in Table 4 is, for each value of E,/G,2. the maximum angle between the isotropic and anisotropic directions for each one of the three principal stress field components as Et/E 2 varies between 0.5 and 2. This angle does not exceed 26 ° . In view of the general form of equation (12) and for a fixed orientation of the planes of transverse isotropy with respect to the CSIRO cell, the in situ stress field components depend on the value of E, and the five parameters of equation (8). A question that arises is " H o w important is the ratio E/Et in comparison to E, when measuring the in situ state of stress with the CSIRO cell?" In order to answer this question, consider again planes of transverse isotropy striking in the north-south direction and dipping at 30 ° to the east.

AMADEI:

335

STRESS MEASUREMENTS IN ANISOTROPIC ROCK 9 QI

%

--w=O °

c a3

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= ~o

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90"

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•

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Fig. 7. (a) Variation o f o j , crz, o 3 with the strike angle ~ w h e n ~b = 0 °, 30 °, 90 °, E / E I = 0.1; E I / E ~ = 0.5; E,/G,2 = 4.38; v2~ = 0.27; v32 = 0.25. (b) Orientation of aj, o2, o 3. Lower hemisphere stereographic projection.

336

,~MADEI:

STRESS MEASUREMENTS

IN A N I S O T R O P I C

ROCK

T a b l e 3. Errors induced by neglecting rock anisotropy. E~.G~: = 4.38: v:~ = 0.27: v~: = 0.25: E k., = 0 I ¢,,

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' M a x i m u m relative error on principal stress m a g n i t u d e in o . ~ M a x i m u m angle between isotropic and a n i s o t r o p i c principal stress directions.

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8 75

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Figure 9 shows the variation of the ratios ch/E ~, t~:/E~ and oa/E, with Et/E: for the three values of E/E, considered in this study. E,/G~: is equal to 4.38. The isotropic solution is also shown for comparison. It appears that for any fixed value of the ratio E,/E:, the magnitudes of the principal m situ stress field components are almost proportional to E~ and their orientation is only slightly affected by the value of the ratio E/E~ as for the isotropic solution shown in Table 2. Indeed. the value of the ratio ot/E I, a:/E~ and a~IE~ does not vary much with E/E.. This conclusion has also been reached when E~/G~, is equal to 2.19 and 8.75 and for other values of the strike and dip angles. As mentioned at the beginning of this section, a complete proportionality of the magnitude of a,, 0:, 03 with respect to E~ would take place if the strain gauges in the CSIRO cell were positioned directly on the pilot hole rock surface with the Young's modulus of the cell material taken equal to zero and with a ratio outer/inner radii equal to 1. Figure 9 also shows how g~/E~, a:/E~ and o3/E ] would vary with E~/E: for that special case. The small variations of the curves in Fig. 9 around the "'no inclusion" solution are associated with the 3 mm thickness of the CSIRO cell and the fact that the embedded strain gages are located 1.5 mm away from the pilot hole wall. In view of the previous conclusion, the following practical questions arises, " H o w large an error is involved by neglecting the geometry and the properties of the CSIRO cell by assuming the strain gauges to be glued on the pilot hole when determining the in situ stress field?" For the example considered in Fig. 9, the relative error involved in neglecting the CSIRO cell varies between 1 and 8°'0 for 0~, i and 13~o for 0, and 2 and 18° 0 for the three values of E/Ej and Et/E 2 considered in the numerical example. The directions of the principal stress field components are only a few degrees apart to those obtained when considering the cell. Although Fig. 9 was obtained for fixed values of Et/G~2 and the angles fl, ~k, the relative errors mentioned above seem to apply also for the other values of the orientation angles and E,/G~. CONCLUSIONS

E~/E2 Fig. 8. (a) V a r i a t i o n o f a~ with the ratio E~,/E 2 for different values o f El,'Gp. when fl = 0 ' , ~ = 3 0 - ; v2~ = 0 . 2 7 : v 3 : = 0 . 2 5 ; E , ; E I = 0 . 1 . (b) C o r r e s p o n d i n g v a r i a t i o n o f a,. (c) C o r r e s p o n d i n g v a r i a t i o n of try.

The measurements of strain or displacement recorded during overcoring in anisotropic rocks such as schists, slates, gneisses, sandstones or shales with the CSIRO cell

AMADEI: STRESS MEASUREMENTS IN ANISOTROPIC ROCK

337

Table 4. Errors induced by neglecting rock anisotropy. ~ = 0:: @= 30:, v,t = 0.27; v3.,= 0.25; E/E~ = 0.1 Angleb Angle~ Angleb Error~ () Error' (:) Error" C) 0.5 35 36 35 2.19 1 6 10 5 5 10 9 2 20 39 60 0.5 30 23 61 4.38 1 21 23 17 17 32 20 2 63 69 11 0.5 14 6 42 8.75 I 46 26 51 11 34 24 2 116 111 20 "Relative error on principal stress magnitude in %. bMaximum angle between isotropic and anisotropic principal stress directions.

EjGI,

Et/E:

(a)

v

E/E~" 0 0 5

o

E/E.~'O I

~' E/E,,." 0 4 5

%

T

:i 2L L 0

I

I

I

I

05

'I0

15

20

E,/E

2

(b)

~'0

0

I

,

I

05

10

15

20

I

I O5

I 10

1

15

I 2O

(c)

"o

0

E~IE z

Fig. 9. (a) Variation of al/E~ with the ratio Et/E2 for different values of E/EI. fl =0°; ~O= 30=; EJGI,=4.38; v21=0.27; h2=0.25. (b) Corresponding variation of tr2/E~. (c) Corresponding variation of

~3/£1.

or the USBM gauge can be analyzed in terms of in situ stresses by using the closed-form solutions proposed by Amadei [9] for the equilibrium of an infinite anisotropic medium bounded internally by an isotropic inclusion and loaded at infinity by a 3-D stress field. For the USBM gauge, this can be done by assuming that the overcoring diameter is at infinity as long as the gauge is positioned directly on the walls of a pilot hole. If the gauge is inserted into a hollow inclusion that has already been bonded on a pilot hole walls or if the C S I R O cell is used, neglecting the finite character of the overcoring diameter and using the closed form solutions is only valid if the residual stresses induced in the inclusion or in the cell and in the overcored sample are within a region near the contact rock, inclusion or rock, cell. In addition, the residual stresses must remain small in order to avoid any separation along that contact [12]. Another expression for these closed form solutions has been proposed by Amadei [12] and has been summarized herein. This enables a better assessment of the influence of rock anisotropy on in situ stress measurements and the error involved by neglecting rock anisotropy. The computation of rock in situ stresses with the C S I R O cell or the USBM gauge depends on several parameters such as (i) the anisotropic character of the rock, (ii) the degree and type of rock anisotropy, (iii) the orientation of anisotropy with respect to the instrumented devices. In addition, when using a C S I R O cell or a USBM gauge that is inserted into a hollow inclusion, the computation of the in situ rock stresses depends also on the relative deformability of the rock with respect to the deformability of the cell or inclusion material This is expressed by the ratio E/Et for an isotropic rock; E/E,, EJEz, EJGI, for a transversely isotropic rock; and, E/E~, EJE2, EI/E3, EJGI2, El/Gl3, EI/Gz3 for an orthotropic rock. When the USBM gauge is positioned directly on the walls of a pilot hole, the dependency on E/E~ vanishes. Neglecting anisotropy by assuming that the rock is isotropic can create large errors when calculating the magnitude and orientation of the in situ stress field. A numerical example dealing with a rock modelled as a transversely isotropic material in which in situ stresses are measured with a C S I R O cell has shown that, for a fixed value of the ratio E/E~, the errors depend on the

338

AMADEI:

STRESS M E A S U R E M E N T S IN ANISOTROPIC ROCK

orientation of the planes of transverse isotropy with respect to the cell and on the values of the ratios E~/E: and EdGe,. Errors as large as I I00o for the magnitude of the calculated principal in situ stress components and orientation 50 ~ off from the isotropic solution have been found in the example presented herein. It has also been found that, for a fixed orientation of the planes of transverse isotropy and for a fixed degree of rock anisotropy defined by the ratios E,,"E2 and E/G,2, the magnitude of the principal in situ stress components is almost proportional to E~; and their orientation is almost unaffected by the value of the ratio E,.'Ej. A perfect proportionality and independency from E/E~ would take place if the strain gauges were positioned on the pilot hole walls and the geometry and Young's modulus of the material of the cell were to vanish. The error involved in neglecting the geometry and properties of the CSIRO cell when measuring in situ stresses, by assuming the strain gauges to be glued directly on the walls of the pilot hole. was found not to exceed 20'!o for the magnitude of the principal in situ stress components and a few degrees for their orientation. This was found to apply for both isotropic and transversely isotropic rocks. This conclusion suggests that, when using the CSIRO cell. the ratio E,,'E~ does not carry as much weight as E~ and the degree of rock anisotropy for the in situ stress determination. Acknowledgements--The author gratefully acknowledges the support of Dr Barry Brady for providing the opportunity to work at the CSIRO Division of Geomechanics in Melbourne (Australia) dunng the summer of 1983. That work contributed to the crystallization of the ideas in this paper.

Received 3 Januao' 1984; revised 25 April 1984.

REFERENCES 1. Goodman R. E. Introduction to Rock Mechanics. Wiley, New York (1980). 2. Berry D. S. and Fairhurst C. Influence of rock anistropy and time dependent deformation on the stress relief and high modulus inclusion techniques of in situ stress determination. In Testing Techniques for Rock Mechanics, pp. 190-206. ASTM, STP 402 (1966).

3. Berry D. S. The theory of stress determination b.~ means of sires, relief techniques in transversely isotropic medium. Tech. Rept N o MRD 5-68. Missouri River Division, Corps of Engineers. Omaha. Nebraska (I 968). 4. Becker R. M. and Hooker V. E Some anisotroplc considerations, in rock stress determinations. USBM RI 6965 (19671 5. Becker R. M. An anisotropic elastic solution l\'~r testing stress rehe( cores. USBM RI 7143 (19681. 6. Berry D. S. The theor? of determinauon ol stress changes in a transversely isotropic medium usmg an instrumented cylindrical inclusion. Tech. Rept No. MRD 1-70. Missouri River Division, Corps of Engineers, Omaha. Nebraska (1970~. 7. Nieva Y. and Hirashima K. The theor~ ot the determination of stress in an anisotropic elastic medium using an instrumented cylindrical inclusion. Mere. Fat'. Engng Kvoto Univ. 33, 221. 232 (197[). 8. Hirashima K. and Koga A Determination of stresses m anisotropic elastic medium unaffected by boreholes from measured strains or deformations. Proc. Int. Syrup. on Field Measurement m Rock Mechanics (Edited by K. Kovari). Vo[. 1, pp. 173-182 (1977) 9. Amadei B. The influence of rock anisotropy on measurement o1 stresses in situ. Ph.D. thesis. Univ. of California, Berkeley (19821. Also published by Springer Verlag in Lecture Notes In Engineering series under the title Rock Anisotropy and the Theory o f Stress" Measurements ( 1983 ). 10. Worotnicki G. and Walton R. J. Triaxial hollow inclusion gauges for the determination of rock stress in situ. Proc. I S R M Syrup. on Investigation o f Stress in Rock and Advances in Stress Measurement. Supplement, pp. I-8, Sydney (1976). I1. Merrill R. H. Three component borehole deformation gage for determining the stress in rock. USBM Rl 7015 (1967). 12. Amadei B. Applicability of the theory of hollow inclusions for overconng stress measurements in rock. Rock Mech. Rock Engng (1984). Submitted. 13. Draper N. R. and Smith H. Applied Regression Analysis. Wile). New York (1966). 14. Amadei B and Walton R. J. Analysis of data obtained with the CSIRO cell in anisotropic rock masses. CSIRO rept (in preparation). 15. Amadei B. Number of boreholes to measure the state of stress in sttu. Proc. 24th U.S. Rock Syrup., Texas A & M Univ.. pp. 87-98 (1983). 16. Fitzpatrick J. Biaxial device for determimng the modulus of elasticity of stress relief cores. USBM RI 6128 (1962). 17. Walton R. J. Mini user's manual for program NMP5 and method of analyzing CSIRO HI cell test data. CSIRO internal rept No. 28 ( 1t~80). 18. Leeman E. R. and Hayes D. J. A technique for determining the complete state of stress in rock using a single borehole. Proc. Ist Congr. I S R M Lisbon, Vol. II, pp. 17-24 (1966). [9. Pickering D. J. Anisotropic elastic parameters for soils. Geotechnique 20. 271-276 (1970). 20. Walton R. J. and Worotnicki G. Rock stress measurements in the 18CC,'12C22 crown pillar area of the CSA mine, Cobar, NSW. CSIRO project rept No. 38 (1978).