Pure appl. geophys. 150 (1997) 203–2150033–4553/97/020203–13 $ 1.50+0.20/0

A Theoretical Model for Kaiser Effect in Rock

C. A. TANG,1,2 Z. H. CHEN,1,2 X. H. XU,1,2 and C. LI3

Abstract—The mechanism of Kaiser effect was studied with the aid of a damage model for rock.Recognizing that the AE counts are transient elastic waves due to local damage of the rock, thequantitative relation between AE counts and statistical distribution of the local strength of the rock hasbeen established. Subsequently, according to Damage Theory, an expression for Kaiser Effect underuniaxial stress state was derived from the model. This is found to be in good agreement with theexperimental results.

Key words: Kaiser effect, acoustic emission, rock failure, damage, seismicity, non-linearity.

1. Introduction

Although Kaiser effect was first observed in metals, there are a number ofpotential uses in geomaterials as an indicator of prior stress conditions. As weknow, the stress in the earth’s crust can rise or fall owing to the influence ofrepeated tectonic motions, or the readjustment of stress fields before and after largeearthquakes. A thorough experimental investigation has been carried out on Kaisereffect by LI and NORLUND (1993a). Presently, the effect has been used to determinein situ stress, assess damage in rock and study the seismicity behavior prior to anatural earthquake (KANAKAWA et al., 1981: HUGHSON and CRAWFORD, 1987;HOLCOMB and COSTIN, 1986; LI and NORLUND, 1993b; MOGI, 1985). However,the bulk of the work only relates to laboratory studies, and as pointed out byHOLCOMB (1993), ‘‘there has been little or no development of a theory of Kaisereffect, particularly for rocks.’’ The purpose of this paper is to develop a model forKaiser effect based on a theoretical link between damage and acoustic emission(AE) in rock. The connection to the Kaiser effect is the assumption that localdamage in rock produces AE. To achieve this, we begin with a model of the growthof local damage in rock under laboratory stress states.

1 Center for Rockbursts and Induced Seismicity Research, Northeastern University, P. R. of China.2 Non-linear Mechanics Laboratory, China Academy of Science, P.R. of China.3 Lulea University of Technology, Lulea, Sweden.

C. A. Tang et al.204 Pure appl. geophys.,

2. Theoretical Link between Damage Parameter D and AE counts N

The earth’s crust contains numerous weak zones, such as existing faults, and isby no means mechanically homogeneous. If the stress level rises, small fractures andslips are likely to occur at these weak zones. This is the so-called ‘‘continuousdamage’’ which means that the material is damaged by the microfracturing contin-uously from the early stage of low stress levels to the end stage of a macro-fault. Bydefinition, we know that the AE are transient elastic waves generated by the rapidreleases of elastic energy due to local microrupturing as the results of local damage.Therefore, the so-called AE are in fact the elastic waves generated by such localdamage radiating through the surrounding material. It is believed that there mustbe a strong link between the rock damage and the AE. In other words, the AEactivity indicates the extent of local damage in rock, which is directly associatedwith the evolution and propagation of fractures within rock.

A theory concerning this developed by TANG and XU (1990), based on DamageMechanics was used here. A brief review of the theory follows.

We can divide the cross section of a rock specimen into many micro-elements asshown in Figure 1. Because each element contains unequal numbers of defects,these elements must possess different strengths. Assume that local strengths, os

(measured in strain), are distributed following a certain probability distribution. Asa load is applied to the specimen, the strain o (presumedly being distributeduniformly) is progressively increased, and individual elements fail locally when o

exceeds the local strength os, i.e., when o\os.Thus we assume:

(1) The local elements of rock specimen are linearly elastic (the macroscopicbehavior of rock may not be linearly elastic), that is:

Figure 1A sketch showing local elements in a cross section of rock specimen.

Theoretical Model for Kaiser Effect 205Vol. 150, 1997

si=oi · ei (i=1, 2, . . . ), (1%)

where ei and si are the Young’s modulus and the stress of the ith element. Weassume that the strains, oi, are uniformly distributed, and the elastic moduli, ei

for elements are the same, then we have

s=o · e. (1)

(2) The strengths of local elements follow certain statistic distribution,

8=8(os ), (2)

where os is the strength of the individual element (local strength).Thus, as the strain of a local element increases, it will fail locally when its strain

exceeds the local strength. Then the proportion of locally damaged elementsmeasured in the area is equivalent to the probability that o\os. Therefore, thedamaged area in the cross section of the specimen is

S=Sm

& o

0

8(x) · dx, (3)

where Sm is the area of overall cross section. Consequently, we can obtain thefollowing expression for the damage parameter D (KRAJCINOVIC and SILVA, 1982)

D=S

Sm

=& o

0

8(x) · dx. (4)

Assume the AE counts in one unit area of the damaged local element are n, thenthe damage in area S will give accumulative AE counts N as the following

DN=n · DS. (5)

If the accumulative AE counts when Sm is completely damaged are Nm, equation (5)can be rewritten as

DN=Nm ·DSSm

. (6)

When a strain increment is applied to the specimen, the increment of the resultantdamaged area will be

DS=Sm · 8(o) · Do. (7)

Substituting equation (7) into equation (6), we have

DN=Nm · 8(o) · Do. (8)

Therefore, the accumulative AE counts, when the strain in the compressed speci-men increases to o, will be

C. A. Tang et al.206 Pure appl. geophys.,

N=Nm

& o

0

8(x) dx (9)

or

NNm

=& o

0

8(x) dx. (10)

This is the accumulative AE counts, N, vs. strain o, when the specimen is loaded.From equations (4) and (10) we can derive the following important relationbetween damage parameter D and AE counts N

D=N

Nm

. (11)

Therefore, 8(o) can also be related to the AE counts of rock. In general, theonset of AE indicates the beginning of further damage in rock (TANG and XU,1990; LI and NORLUND, 1993a,b).

According to damage mechanics (KRAJCINOVIC and SILVA, 1982), the constitu-tive laws of rock under uniaxial stress conditions can be expressed as

s=Eo(1−D), (12)

where s and E are the stress and the elastic modulus of the rock specimen,respectively. Therefore, we can obtain the constitutive law of rock under uniaxialstress condition from equations (4) and (12) as the following

s=Eo(1−D)=Eo�

1−& o

0

8(x) dxn

(13)

s=Eo�

1−N

Nm

�=Eo

�1−

& o

0

8(x) dxn

. (14)

From equation (9) we know that if the probability distribution density of localstrength, 8(o), is known, then the accumulative AE counts, N(o), can be obtaineddirectly. There are many probability distribution density functions that can be used,which is the best depends on the properties of rock. We can choose the appropriateone by comparing it with the experimental results. HUDSON and FAIRHURST (1969)used normal distribution in their simple structure disintegration model to expressthe local elemental strength of rock. However, the most commonly used distribu-tion is Weibull’s distribution which describes very well the experimental data fordistribution of crack length within a rock specimen. Other reasons can also befound for Weibull’s theory of the strength of material (HUDSON and FAIRHURST,1969). Here we assume the local elemental strength of rock follows Weibull’sdistribution law. That is

8(o)=mo0

�o−os

o0

�m−1

exp�−�o−os

o0

�mno]os, (15)

Theoretical Model for Kaiser Effect 207Vol. 150, 1997

Figure 2Theoretical curves for AE counts and stresses vs. strain.

where m is the shape parameter and o0 is a measure of average strength. One of theattractive aspects of the Weibull’s distribution is the presence of the shape parame-ter which allows this function to take a wide variety of shapes. For example, form=1, this distribution is exponential; at about m=1.5, the distribution is nearlylognormal; and at about m=4, it very closely approximates a normal distribution.Since the shape parameter m is a measure of the local strength variability, it can beconsidered as a homogeneity index (TANG and XU, 1990). The larger the index m,the more homogeneous is the rock. When m trends to infinity, the variance trendsto zero and a ‘‘perfect’’ rock is obtained. A material with such a property is theso-called ideal brittle material, such as glass.

Considering the random distribution of strength of local elements, equation (15)is multiplied with a factor, RND, which has a random value between 0 and 1generated by a computer, that is:

8(o)=RND(o) ·mo0

�o−os

o0

�m−1

exp�−�o−os

o0

�mno]os. (16)

Four stress-strain and AE counts-strain curves (i.e., s−o, 8−o) are shown inFigure 2 by using equations (14–16) with four different homogeneity indices. Thenonlinear behavior occurring at the stress-strain curves is caused by the localdamage in the rock. It shows that failure progresses (following peak stress) asaverage applied stress decreases. That is to say, even if the maximum strain strength

C. A. Tang et al.208 Pure appl. geophys.,

of the sample is exceeded, the micro-failure continues to occur. This is the so-calledpost-shock, a term often mentioned in seismology.

Figure 3 supplies a comparison between an experimental and a theoreticalresult. The tested specimen is granite. The theoretical result is found to be in a goodagreement with the experimental result, which also shows that there is goodcorrelation between the stress-strain characteristics and the AE counts during theloading history of the specimen. The results reveal that the strong correlationbetween AE activity and inelastic strain can be used to measure damage accumula-tion in brittle rocks.

Figure 3The stress-strain curve and the AE record for granite specimen. (a) The experimental data, (b) the

simulations.

Theoretical Model for Kaiser Effect 209Vol. 150, 1997

3. Kaiser Effect of Rock

We know from the above-mentioned definition of the density function of thelocal strength distribution that equations (2), (15) or (16) show irreversibility ofdamage with the strain variations. The increase of deformation is accompanied bythe damage of local elements. Obviously, the strength of those damaged localelements is lost and unrecoverable even if the deformation may be restored, or theload is removed. As mentioned by LOCKNER (1993) that if the stress-induceddamage (microcrack growth) in a rock is irreversible then it is likely that significantnew damage will occur only when the previous stress state is exceeded. This isessentially the rationale for the Kaiser effect and has led HOLCOMB and COSTIN

(1986) to use AE as a technique for probing stress space to determine themagnitude and shape of the damage surface for a rock. Therefore, if the reloadingstrain does not exceed the previous level, the increment of the damaged area of thelocal elements should be zero, DD=0. No damage means no acoustic emission.Nevertheless, when the reloading strain exceeds the previous level, new damage willbe created and AE emits again. Hence the reloading strain will be related to theaccumulative AE counts as the following

N2

Nm

=!0

oo 1

8(x) dxo5o1

o\o1

, (17)

where the subscript denotes the number of loading cycles and o1 stands for thestrain level at the previous loading.

For the situation of repeated loading of the ith time, we have the generalizedequation of the accumulative AE counts as the following:

Ni

Nm

=!0

ooi−1

8(x) dxo5ooi−1

o\ooi−1

(18)

in which i=1, 2,3 . . . and oi−1 denotes the strain level at the (i−1)th loading.Equation (18) is the general function of Kaiser effect derived in this paper from

the statistic approach to the rock failure.The theoretical graphs of equation (18) when 8(o) takes the Weibull’s distribu-

tion law have been plotted in Figure 4 for a homogeneity index m=2. During thefirst loading, AE occurs due to the local damage of rock. No further AE occurs inthe process of unloading. During reloading, the growth of local damage does notoccur until the stress reaches the previous maximum stress level. Then, growth oflocal damage and the accompanying AE resume.

Figure 5 displays the experimental graphs for a granite specimen under progressiveloading. The specimen was loaded and unloaded eight times before it failed. At thebeginning of the first loading cycle, numerous AE occurred, which is a commonphenomenon in the test of rock specimens. In the second loading cycle, AE was absentuntil the load reached the maximum stress level of the first cycle. At that point, thecontinuation of AE resumed. Similar phenomena occurred in the subsequent cycles.

C. A. Tang et al.210 Pure appl. geophys.,

Figure 4Theoretical curves of AE counts under repeated loading conditions, m=2.

By comparing Figure 4 with Figure 5, it can be seen that when considering thetrends, the theoretical result agrees quite well with the experimental results exceptat the beginning of the first cycle.

The AE counts can also be plotted in an accumulative form. As a comparison,the simulation and experimental results of AE shown in Figures 4 and 5 arereplotted in Figures 6 and 7 in an accumulative form, respectively.

4. Discussion and Conclusion

Although great attention has been paid to the study of Kaiser effect both inmodel experiments and observations during natural earthquakes in recent years,few convincing quantitative explanations have been given to the results observed.

Using the arguments presented in this paper, it is possible to offer a quantitativeexplanation for Kaiser effect from the viewpoint of damage mechanics of rock.Although it considers only the uniaxial case, the model may be extended to casesunder confining pressure. The establishment of this model supplies a possiblequantitative explanation of the AE and the Kaiser effect. It might be used as aguide for potential applications of the Kaiser effect, for instance, explanations of

Theoretical Model for Kaiser Effect 211Vol. 150, 1997

experimental data of AE and field observations of natural earthquakes, assessmentof damage, etc.

The main problem with the analytical method proposed in this paper is that theresults are generated under the assumption of independent noninteraction damageevents. This assumption is clearly violated once fracture localization occurs. Sincethe theoretical prediction (Fig. 6) does not include the interaction effect, itunderestimates the cumulative AE near peak stress. Also, the time-dependent crackgrowth is not taken into account here.

Figure 5The experimental records of the AE counts of a Zinkgruvan ironore specimen under uniaxial cycle

loading (the uniaxial compressive strength=236 MPa).

C. A. Tang et al.212 Pure appl. geophys.,

Figure 6Theoretical curves of accumulative AE counts vs. Strain under repeated loading conditions.

Figure 7The accumulative AE counts of Zinkgruvan ironore specimen under uniaxial cycle loading.

At the moment, we are focusing on the numerical simulation pertaining to rockfailure-induced acoustic emissions or seismicities. In this way, the interaction effectof damage events will be taken into account and the failure localization is modeled.

Figure 8 is one of the examples of the numerical simulation on interactiondamage events and the localization. Due to the space limitation, only a few steps

Theoretical Model for Kaiser Effect 213Vol. 150, 1997

Figure 8Numerical simulation of rock failure and induced seismicity.

C. A. Tang et al.214 Pure appl. geophys.,

are presented in this paper. The full description of the numerical simulation methodwill be published elsewhere (TANG, 1997). The simulation result shows that, duringthe initial loading phase (low stress level), a few fractures are localized andrelatively sparse (see step 18 in Fig. 8). As microfracture damage accumulates,fractures become clustered, involving more elements, leading to fracture interaction(step 20 in Fig. 8). As more damage accumulates, the number of failed elementsincreases drastically as shown in the central area. The process becomes mechani-cally unstable, i.e., the central area in the weaker zone suddenly collapses forminga fault (step 23). As a result, the fault produces its own stress field which producesfurther microfracture damage around and ahead of its tips (step 25). This is theso-called post-shock.

It is seen from Figure 8 that the amount of damage occurring in the rock duringfailure, and hence the possibility to rupture, can be predicted better by methodswhich accommodate the progressive failure and associated seismic activities. Thecombined use of failure event modeling and event source location determinations,in conjunction with deformation simulation, will provide additional insight into theproblem. The current development efforts focus on the Kaiser effect modeling innumerical method.

Acknowledgements

The support for this research through the National Natural Science Foundationof China and China National Education Commission is gratefully acknowledged.Comments of an anonymous reviewer are appreciated.

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(Received May 3, 1996, accepted April 2, 1997)

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