Rock Mechanics 7, 231--241 (1975) © by Springer-Verlag 1975

Technique and Apparatus for Strain Measurements on Rock in Constant Confining Pressure Experiments By W. R. W a w e r s i k With 4 Figures (Received July 9, 1974) Summary - - Zusammenfassung

- - R6sum~

Technique and Apparatus [or Strain Measurements on Rock in Constant Confining Pressure Experiments. An indirect scheme to measure radial strains on competent rock cylinders and joint deformations in jointed specimens subjected to conventional triaxial compression is described. The technique was first employed by C r o u c h (1970). It relies on the measurement of the volume changes of the confining pressure medium that are necessary to maintain a constant confining pressure during radial sample deformation. In the simplest case when the cross-sectional areas of the loading pistons and of the test sample are equal it is shown that radial strain is given by an expression of the form s2 = ~

1

[L--XAC~V -

(C2 ~1 + Ca) F]

where SA V denotes cumulative volume adjustments of the confining pressure fluid. Similar equations in some cases define the effective normal and shear displacements on single joints. Equipment and procedures are described by which radial strains and joint displacements were monitored in quasi-static compression tests and in creep experiments up to 10,000 psi confining pressure.

Untersucbungsver[ahren und Gerh'te [~ir Ver[ormungsmessungen an Gesteinen bei Versuchen mit gleichbleibendem Manteldruck. Beschrieben wird ein indirektes Verfahren zur Messung yon Radialverformungen fester Gesteinszylinder und yon Kluftverschiebungen in geklfifteten Probek6rpern w~ihrend herkSmmlicher Triaxialversuche. Das Verfahren wurde zuerst yon C r o u c h (1970) angewendet. Es beruht auf der Messung yon Volumen~inderungen der Manteldruckfliissigkeit die notwendig sind, um den Manteldruck konstant zu halten. Es wird gezeigt, daf~ im einfachsten Fall, wenn die Querschnitte der Belastungskolben und der Proben gleich sind, die Radialverformung dutch die Formel 62=~

1

[S, A V - ( C 2

L C1

el+Cz)

F]

ausgedriickt werden kann, wobei Z'L]V die Volumsanpassung der Manteldruck-

232

W.R. W a w e r s i k :

fliissigkeit bezeichnet. Khnliche Gleichungen ergeben in bestimmten Ffillen die mittlere Normal- und Scherverformung an einzelnen Kliiften. Es werden Get,ire und Verfahren beschrieben, bei denen Radialverformungen und Kluftverschiebungen bei quasistatischen Druck- und Kriechversuchen bis zu 10 000 psi Manteldruck verfolgt wurden. Technique et instrumentation utilisdes pour Ies mesures de ddformation de roches dans des expdriences ~ pression de confinement constant. Une mSthode in-

directe est d&rite pour les mesures de dfiformation radiale dans des cylindres de roche comp&ente et des d~formations de joint dans des sp~cimens jointifs sujets une compression triaxiale conventionelle. Cette technique fur utilisfie pour la premiere lois par C r o u c h en 1970. Elle est basde sur la mesure des variations de volume du fluide pressurisant n~cessaire ~ maintenir une pression de confinement constante durant la dfiformation radiale du spdcimen. Dans le plus simple des cas lorsque les sections des pistons de charge et de l'dchantillon sont identiques, la ddformation radiale est dficrite par une expression de la forme e2 = ~

~

- (C2 e~ + C3)

ou Z d V repr&ente les adjustements cumul6s du volume de fluide pressurisant. Des dquations semblables d4finissent dans certains cas les ddplacements effectifs en tension et cisaillement sur des joints isolds. Description est faite de l'instrumentation et re&bode utilisdes pour les mesures des dfformations radiales et des d@lacements de joints dans des tests de compression quasi-statique et dans des exp4riences de ddformation lente jusqu'h des pressions de confinement de 10000 psi. Introduction A variety of techniques are utilized to measure strains on rock in constant confining pressure experiments. Almost all of them are performed on cylindrical specimens and, for complete strain resolution, require that the strain fields be homogeneous throughout the sample. By far the most common approach relies on the application of SR-4 resistance strain gauges. The strain gauges are bonded either directly to a specimen or to a thin jacket which fits tightly around the sample. Although reliable in many situations the selection of strain gauges to monitor strain in confining pressure tests has several distinct disadvantages: (1) The cost for materials and labor are high and the set-up times of individual experiments are relatively long, (2) strain gauge readings are spurious when rock deformation is accompanied by microcracking, (3) it is often difficult to mount strain gauges on very porous or rough rock surfaces and, (4) strain gauge bonds are unreliable on moist surfaces. Alternative approaches that avoid several of the above problems permit strains in rock to be calculated from measured axial and radial and/or circumferential displacements. Specifically, radial and circumferential displacements are sometimes monitored by means of one or several sets of strain gauged cantilever devices or by means of D C D T or potentiometer transducers. The scheme of strain measurement described in this paper chooses

Technique and Apparatus for Strain Measurements on Rock

233

the same basic approach but is believed to be more accurate and versatile. It was first referred to by C r o u c h (1970). Since that time it has been automated and its scope of application extended. For this reason a more detailed account of the technique is now deemed appropriate. The Principle In a constant confining pressure experiment (o1>o2=oa) typically a cylindrical rock specimen is placed between two loading pistons in a fluid filled pressure vessel. The sample is jacketed to isolate the rock from the surrounding confining pressure fluid. The specimen is then pressurized circumferentially (or hydrostatically) by increasing the pressure of the confining pressure fluid. Subsequently, the axial stress is raised by applying a force to the loading pistons. As the sample is stressed, it deforms both axially and radially and, if the amount of fluid in the confining pressure system is held constant, a change of the confining pressure occurs. In order to maintain the confining pressure fixed the volume of the confining pressure fluid must be adjusted. Considering this series of events more closely it becomes apparent that the volume adjustments required to maintain a constant confining pressure o2 =oa are related to the (average) radial specimen strain. Explicitly, for homogeneous sample deformation it can be shown that e8 = s2 = ~

1

[ Z cd~VT -

-

(1)

(C281-{-C3) ~ q - C 4 @I-q-CsF)]

where

C1 =2 C2

AtLt

C4 = (A~-A~) 2At

~'s

C5

GA~

Lc E, At Lt

L~

The symbols in Eq. (1) denote the following: = cross sectional area of test sample and end-caps that are commonly placed between the specimen ends and the loading pistons. length of test sample. gt combined length of end-caps. Le elastic constants of loading piston and end-cap material. Es, qJ8 As = cross sectional area of loading pistons. LI~ = effective internal length of pressure vessel. F = axial force; and X A V = cumulative, incremental volume adjustments of confining pressure medium. At

Rock Mechanics, VoL 7/4

17

234

W.R. W a w e r s i k :

The derivation of Eq. (1) is based on the assumptions that the temperature of the confining pressure fluid remains constant and that the radial strain e2 is uniform and uneffected by end constraints. In reality frictional end constraints at the sample-loading piston (end-cap) interfaces exist resulting in smaller strains near the ends than at the center of the specimen. Therefore, the average radial strain e~ that is calculated from Eq. (1) establishes only a lower bound for the radial strain at a distance Ld2 from the specimen ends. However, calibration tests have demonstrated that the difference between the average and true strain is small if the length-to-diameter ratio L : D of the sample is not less than 2.5 : 1 and if the axial strain el does not exceed

t COMPETENT SAMPLE (C.S.)

(oi

JOINTED SAMPLE (J.S.)

-o3)' ///~C.s.

(G)

Fig. 1. Schematic comparison of stress-strain response of jointed and competent rock Schematischer Vergleich des Spannungs-Dehnungsverhakens yon gekli.iftetem und festem Gestein Comparaison sch~matique des charactdristiques contrainte-d4formation de roches jointes et

comp&entes twenty per cent. Length-to-diameter ratios of two can be tolerated for specimen diameters D~
1

(c

l+c3)F]

(2)

Technique and Apparatus for Strain Measurements on Rock

235

If the diameters of the rock specimen and loading pistons are not equal, Eq. (2) should be used only after the last term in Eq. (1) was proven to be negligible. For small strain el Eq. (2) reduces to (3) below s2=e3=

XA V

c~-C~F

(3)

Before Eqs. (1) through (3) are applied to any system several calibration tests must be carried out. For matched diameters of rock specimen and loading pistons the calibration of the radial deformation system generally requires experiments to be performed on three standard samples of known elastic moduli and Poisson's ratios. Two calibration experiments may suffice if the elastic constants of the loading pistons are known. In that event one test is used to verify that C l = 2 A t L t while the other serves to ascertain the effective value of L v including the unpredictable influence of standard O-ring seals. If twice the radial strain e2 is interpreted as the lateral component of the volumetric strain of the test specimen ( C r o u c h , 1970) then the technique described above can be employed, in some cases, to determine the average normal and shear displacements d~ and ds along discrete joints in rock. If the stress-strain data for competent rock samples are reproducible, the displacements d~ and ds can be calculated from comparative measurements of the average axial and volumetric strain of jointed and competent samples. To demonstrate this point, in Fig. 1 principal (applied) stress difference is plotted versus strain. Strain here denotes either volume strain or axial strain. Clearly, the difference in strain between the jointed and the competent sample A e at any stress level must be due to the joint. From this difference the average normal deformation on a single joint is derived as dn = vaj~ A e

(4)

where Vt = sample volume, and aj = joint area; Ae = difference in volumetric strain between jointed and competent specimens; d~ > 0 for joint closure. Once d~ is known the average joint shear displacement d8 is given by the equation d8 = A e~ L t - dn sin c~

(5)

COS ¢g

where Lt = specimen length; c~ = (smallest) angle between slip direction and specimen axis; Ael = difference in axial strain between jointed and competent specimens. 17"

236

W.R. W a w e r s i k :

The displacements dn and de can be related to the average normal and shear stresses ~ and rn on the joint, for example to calculate the so-called joint stiffnesses. I ....

PRESSURE

VESSEL

I

" CONTROLCIRCUIT

~L 1 SPEED

I

~

~ PRESSURE REGULATOR-VOLUME METER PRESSURE GAUGE

Abb. 2 a

Abb. 2 b Fig. 2. (a) Schematic of confining pressure and strain monitoring system (b) Pressure regulator/volume meter and servo-control system (a) Schema der Druck- und Verformungsmel~einrichtung (b)

Druckregulator/Volumenmefgapparaturund Servo-Steuersystem

(a) Schdma du syst~me de pressurisation et de mesure de d~formation (b) RSgulateur de pression/dSbit-m&re et syst~me de servo-contrbie

Technique and Apparatus for Strain Measurements on Rock

237

Apparatus and Procedure A new system was designed which is capable of regulating confining pressure in standard confining pressure experiments and of measuring strains according to Eqs. (1) through (3). In principle, this system (Fig. 2) is an updated version of manually controlled equipment used earlier by C r o u c h (1970). T h e main components of the apparatus are (Fig. 2): (1) pressure vessel; (2) motor-driven, servo-controlled, screw type pressure regulator/volume meter; and (3) differential pressure gauge for feedback control of the pressure

Fig. 3. Original record of axial force versus axial and radial strain (displacement) for Tennessee marble subjected to 1,000 psi confining pressure (strain rate el ~ 10-5/sec) Laboraufnahme yon Axiallast gegen Axial- und Radialverformung yon Tennessee-Marmot unter 1000 psi Manteldruck (Verformungsgeschwindigkeit kl ~ 10-5 sec-1) Enregistrement original de la force axiale et des ddformations (dSplacements) axiales et radiales du marbre de Tennessee soumis ~t une pression de confinement de 1000 psi (vitesse de d6formation: kl ~ 10-5/sec) regulator. In its current design the system can sustain a line pressure of 30,000 psi. Volume changes of the confining pressure medium are resolved to within approximately 4 × 10 .4 in a. T o accomplish this the differential pressure gauge senses pressure deviations of less than _+3 psi using a bellows and a built-in L V D T transducer. T o avoid damage to the instrument during uncontrolled brittle rock fracture it is hydraulically "buffered" to withstand overloads of _+3,000 psi.

238

W.R. Wawersik:

6.0

- ~

P = 3 , 0 ~ psi.

1

I

I

x

T nO Z

W.G., J - 3 3.0,

P= 1,000 psi.

1.5

-8

-4

0

NORMAL

4

DISPLACEMENT,

8 x l 0 -4 in.

Abb. 4 a Fig. 4. (a) Average normal stress a,, versus average normal displacement d~ for rough tension joint in water-saturated Westerly granite cylinder in quasistatic confining pressure experiments (d. > 0 denotes joint closure) (b) Average shear stress T~ versus average shear displacement ds for rough tension joint in water-saturated Westerly granite cylinder in qnasistatic confining pressure experiments (a) Mittlere Normalspannung gegen mittlere Normalverschiebung d . f~ir eine rauhe Zugkluft in einer wassergesfittigten Zylinderprobe von Westerly-Granit im quasi-statischen Manteldruckversuch ( d . > 0 bedeutet eine Klukverengung) (b) Mittlere Schcrspannung gegen mittlere Scherverschiebung d~ ffir eine rauhe Zugkluft in einer wasserges~ittigten Zylinderprobe von Westerly Granit im quasi-statischen Manteldruckversuch (a) Contrainte normale moyenne ~,~ et d6placement normal moyen d . d'un joint de tension rugueux dans un cylindre de granit Westerly saturd d'eau dans le cas d'expdriences ~t pression de confinement quasi-statique (d~ > 0 denote la fermeture du joint) (b) Contrainte de cisaillement moyenne ~,, et dSplacement en cisaillement moyen d~ d'un joint de tension rugueux dans un cylindre de granit Westerly satur~ d'eau dans le cas d'exp~riences ~i pression de confinement quasi-statique

Technique and Apparatus for Strain Measurements on Rock

239

The loading and pressure control procedure of the confining pressure system in Fig. 2 is effected as follows. As confining pressure is applied at the beginning of an experiment, both ports of the differential pressure gauge are open. After the pressure has reached and stabilized at a pre-determined

/

/ / /

/ ~p

% ] I

/ I

co

3

l

~

P=I,O00

ql

I co

W.G, J-3

0~

I

t

I

I

r

I

4

6

8

I0

12

SHEAR DISPLACEMENT, xfO-5 in.

Abb. 4 b value, one side of the differential pressure gauge is isolated by means of a valve (Fig. 2a). The pressure of the fluid entrapped in the isolated (reference) side of the differential pressure gauge now provides a reference pressure that is maintained in the pressure vessel (active) side through the action of the pressure regulator. Incremental fluid adjustments are monitored by means of a potentiometer that contacts the regulator bushing and, indirectly, tracks the linear movement of the regulator piston. Accuracy To verify the validity and to establish the accuracy of the strain measurement technique, several experiments were conducted on brass, aluminum and steel. The specimen lengths were deliberately varied by approximately 20 per cent. The worst discrepancy among measured physical properties determined indirectly as described above and directly by means of direct bond strain gauges was 1.8 per cent of the smallest observed reading. Generally data using both techniques agreed to within better than 1 per cent. Poisson's ratio of steel in the plastic range was determined only indirectly

240

W.R. W a w e r s i k :

as 0.50 after strain gauges began to "peel" and render spurious, excessively large readings. The value v =0.50 may be high because of possible thermally induced pressure changes in the confining pressure fluid as heat was conducted away from the steel undergoing plastic deformation.

Applications The method for the indirect measurement of strains in constant confining pressure experiments was applied in quasi-static and creep experiments on both competent and jointed specimens of several rock types and of artificial salt which underwent 15 per cent (permanent) axial strain. Both dry and water saturated rock specimens were employed. The strain rate in all quasi-static tests was sz~<10-4/sec, in order to remain within the limits of response time of the confining pressure system and to minimize temperature changes in the pressure fluid. Creep tests were carried out over periods of up to 1,100 hours. Several experimental results are shown to demonstrate the quality of data obtained. Fig. 3 is the reproduction of a typical record of force versus axial and radial strain (displacement) for a marble specimen subjected to 1,000 psi confining pressure [average strain rate ~1=0 (10 5)/sec]. The steps in the trace of force versus radial deformation indicate the frequency, resolution and reproducibility of the servo-controlled pressure/volume adjustments of the confining pressure medium in maintaining the pressure constant. No problems were encountered in tracking post-failure deformations. Figs. 4a and 4b show the relationships between the average normal stress ~n and the average shear stress r n on an artificially created tension joint in a granite on one hand and the normal displacements d,~ and ds on the other. Similar results have been obtained for multiply jointed specimens for which the net axial and volumetric strain contributions due to all joints have been ascertained~ The apparatus that was employed to obtain the data in Fig. 3 through 5 can be utilized in other ways. For example, it can be applied to determine porosity changes due to micro-cracking ( C o r n e t , 1972) or to establish joint (fault) dilatancy in conjunction with joint (fault) slip in constant pore pressure experiments. Other application may prove feasible provided that the pressure changes in the fluid medium can be monitored and that the necessary volume adjustments of the confining pressure system can be resolved with existing equipment and instrumentation. Summary A new apparatus for efficient, indirect strain measurement on rock in constant confining pressure experiments is described. Numerous experiments have proven this techniques to be versatile, accurate and economical. Compared with the use of strain gauges the savings in materials and labor are estimated to be at least $ 30 per test. The method is particularly useful

Technique and Apparatus for Strain Measurements on Rock

241

where strain gauges cannot be employed, for example, on wet specimens or when large strains are encountered. In some cases the technique is suitable to determine joint deformations.

Acknowledgements The work that is described in this paper was partially supported by the Advanced Research Projects Agency under contract No. H0220007 and through a grant from Dr. S. S. K i s t l e r , Professor Emeritus, University of Utah. I should like to thank Dr. S. L. C r o u c h for several helpful discussions during the course of this work and for his careful review of the manuscript. References Crouch, S. L.: "Experimental Determination of Volumetric Strains in Failed Rocks", Int. J. Rock. Mech Min. Sci., 7, 6 (1970). Cornet, F., and C. Fairhurst: "Variation of Pore Volume in Disintegrated Rock", Proc. Symposium on Percolation Through Fissured Rock, ISRM, Stuttgart, September (1972). Address of the author: Prof. Dr. W. R. Wawersik, Shock Wave Phenomena Division (Org. 5163), Sandia Laboratories, Albuquerque, NM 87115, U. S. A.