Technical Note

A new approach to estimating geo-stresses from laboratoryKaiser e�ect measurements

Siqing Qin*, Sijing Wang, Hui Long, Jun Liu

Engineering Geomechanics Laboratory, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029,

People's Republic of China

Accepted 23 September 1999

1. Introduction

It is generally considered that in situ geo-stressmeasurements, such as borehole overcoring andhydraulic-fracturing, are the only direct methods ofestimating the in situ stress ®eld. Nevertheless, becausesuch measurements can be di�cult, especially for med-ium-scale or small engineering, their application is con-strained to some degree. Thus, it is helpful to developan economical and reliable geo-stress measurementmethod using laboratory measurements.

Although some engineers have applied the Kaisere�ect to determine in situ stress [1±4], the Kaiser e�ectresults can be doubtful because determining the Kaisere�ect value is di�cult. In view of this, we put forwarda new approach to measure geo-stresses using thewhole stress±strain curve and a damage model. Also,there is explicit recognition that stress is a tensor quan-tity and at least six measurements are required.

2. Memory model of rock stress: damage model

2.1. Principle of geo-stress memory for rock

Sun [5] pointed out that measurement of the geo-stress of rock is not as good as that of the geo-strain.Qin and Li [3] found by acoustic emission (AE) exper-

iments that pre-strain can be established more e�ec-tively than pre-stress. This indicates that we should®rstly determine strain and then compute stress bymeans of the stress±strain relation.

Under the action of load, as microcracks withinrock nucleate, propagate, interact and coalesce,damage is caused during deformation of the rock.Because the damage is irrecoverable, the damagedstate is memorized. If the reloading strain does notsurpass the pre-strain, no new damage is produced,and if it is reached or exceeded, new damage can begenerated. So we can see that the damage producedwill de®nitely a�ect the reloading complete stress±strain curve before and after the pre-strain; in otherwords, the reloading stress±strain curve re¯ects theprevious damage or strain.

It can be also seen from the repeatedly loading andunloading test curve (Fig. 1) that the loading andunloading process of rock is irreversible and is relativeto the previous forced state or deformation history,and the corresponding relation between the stress stateand the strain state is not unique. That is to say, rockassuredly retains a memory of its past deformation his-tory.

2.2. Damage model

According to damage mechanics [6], the constitutivelaw of rock under uniaxial compression can beexpressed as

s � Ee�1ÿD�, �1�

International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1073±1077

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* Corresponding author. Fax: +86-10-620-405-74.

E-mail address: egml@igcas.ac.cn (S. Qin).

where s, e and E are the stress, the strain and the elas-tic modulus of the rock specimen, respectively, and Ddenotes the damage parameter.

By means of the speci®c analysis of the expressionof D [4,7], we obtain

D � 1ÿ exp

"ÿ�eÿ e0em

�m#, �2�

where m is the shape parameter, em is a measure ofaverage strength and e0 is the initial strain value, i.e.the threshold value corresponding to initial damage.This is the Weibull distribution.

When e< e0, D � 0 and when ere0, Dr0. That isto say, only when the reloading strain value surpassesthe threshold strain e0, new damage can be produced,which corresponds to the rapid increase in AE activi-ties, i.e. the occurrence of the Kaiser e�ect. Thus, e0 isthe geo-strain value to which the rock specimen wassubjected once in the direction of testing.

Substitute formula (2) into formula (1) and obtain

s � Ee exp

"ÿ�eÿ e0em

�m#: �3�

The above formula is the memory model of stress(damage model) established by us. With formula (3),

the geo-stress can be determined by solving the e0value.

It should be pointed out that the above analysis isbased on two assumptions: (1) the strain value memor-ized is during the elastic stage, so the interactionamong cracks can be neglected and (2) only one-termstress is memorized in that direction.

2.3. Determination of e0 value

Take the minimum of the square sum of stress devi-ations as an objective function, and use the leastsquare ®tting and the iterative method to determinethe e0 value. The stress value corresponding to e0 onthe experimental stress±strain curve is the geo-stresscomponent to be measured.

3. Experiment

3.1. Testing system

Samples of marble come from bore cores of PD-4#

tunnel, a hydroelectric power station, in Sichuan pro-vince, P.R. China. They are cut to make six-directionspecimens as shown in Fig.2 with the same dimensionof 2.5� 2.5� 7.5 cm3.

The loading system is a MTS-815 servo-controlled

Fig. 1. Deformational behavior of sandstone under cycling uniaxial compressive load (Fig. 2 in Ref. [8]).

S. Qin et al. / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1073±10771074

testing machine and a constant strain rate during load-ing of 10ÿ5/s was used. Acoustic emission activitiesand stress±strain signals are simultaneously collectedand the specimen is loaded until it is completely bro-ken.

3.2. Experimental results

Fig. 3 is a typical AE experimental curve of marblespecimen 6, and Fig. 4 shows an experimental stress±

strain curve and the theoretical ®tted curve for speci-mens 36 and 33. It is seen that the ®tting e�ect (corre-lation coe�cient is 0.987 for specimen 36, 0.974 forspecimen 33 and >0.963 for the other specimens, re-spectively) is very good, which shows that the damagemodel built is comparatively reliable. Some testingresults for the Kaiser e�ect and the damage model arelisted in Table 1.

The principal stress values calculated from Table 1are listed in Table 2.

Fig. 2. Sampling directions.

Table 1

Contrast between the stress values measured by the damage model and those by the Kaiser e�ect

Sample No. 5 11 18 21 22 31 33 36

Direction X Y Z X458Y X458Y X458Z Y458Z Y458Ze0 value by the damage model (�10ÿ2) 0.13 0.054 0.041 0.124 0.10 0.064 0.07 0.08

Stress value by the damage model (MPa) 15.2 7.8 8.0 13.7 12.3 13.6 8.8 14.1

Stress value by Kaiser e�ect (MPa) 12.1 6.15 9.0 11.3 12.3 13.5 10.0 11.5

S. Qin et al. / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1073±1077 1075

Using in situ stress measurements made by the over-coring method near PD-4# tunnel [4], we know thatthe maximum principal stress value is about 18.2(MPa) with a northwestern direction, which is close tothe tested value from the damage model, but far fromthe one by the Kaiser e�ect. This shows that thedamage model method has a higher accuracy becausejudgement is not required to establish the characteristicpoint.

4. Conclusions

The damage model method eliminates the need forjudgement on the characteristic point of the Kaiser

e�ect and has a good theoretical basis. By means ofcomparison analysis with other direct stress measure-ments, we can say that this new method to estimate insitu geo-stresses has initial practicability. We empha-size that the estimation procedure explicitly recognizesthe tensorial nature of the stress ®eld with its six inde-pendent components.

References

[1] Holcomb DJ. General theory of Kaiser e�ect. Int J Rock Mech

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[2] Kanakawa T, Hayashi M, Kitahara Y. Acoustic emission and

overcoring methods for measuring tectonic stress. In: Proc. Int.

Weak Rock, Tokyo, 1981. p. 1205±10.

Fig. 3. AE experimental curve for specimen 6.Fig. 4. Experimental and theoretical curves for specimens 36 and 33.

Table 2

Principal stress values

Method s1 s2 s3

value (MPa) direction (8) dip (8) value (MPa) direction (8) dip (8) value (MPa) direction (8) dip (8)

Damage model 16.44 N85.5W 27.0 10.21 N10.4W ÿ50.1 4.34 N22.5E 35.2

Kaiser e�ect 12.86 N66.1E ÿ34.9 8.15 N25.8W ÿ55.5 5.08 N11.9E 41.3

S. Qin et al. / International Journal of Rock Mechanics and Mining Sciences 36 (1999) 1073±10771076

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