Sustained acoustic emissions following tensile crack propagation in a crystalline rock

Abstract

We show that simple breakage of a crystalline rock (gabbro) in tension begets further breakage of rock in the area around the first crack that is self-sustaining and spontaneous and that is detected via sustained acoustic emissions (AE). The result is a sequence of AE events that is statistically similar to aftershocks from earthquakes, that scales with the size of the main crack, and that we were able to observe for days following the initial breakage in laboratory-scale experiments. A new model for aftershock generation that is based on residual strain relaxation is shown to be consistent with the observed hyperbolic decay of the event rate with time and with the manner in which the decay law scales with the size of the main rupture.

Appendices

Appendix 1: Additional experimental data

The main article presents a selection of the experimental data. In its support, we here provide a compendium of data from the experiments.

Figure 9 gives the evolution of injection pressure (hydraulic fracturing experiment) or loading force (notched beam experiments) along with cumulative AE events for the experiments not included in the main article. Consistent with the data in the main article, we observe AE prior to peak pressure, during hydraulic fracture propagation, and after fracture propagation ceases with an increase in aftershock frequency coming after the injection ceases and the fluid pressure is released.

Fig. 9

Evolution of injection pressure (hydraulic fracturing experiment) or loading force (notched beam experiments) along with cumulative AE events for the experiments not included in the main body of the paper. For clarity, only the first 6000 s after peak pressure/load is shown. Open circles superimposed on the pressure record correspond to AE events and the vertical dashed line corresponds to shut-in (isolation from the injection system) for the hydraulic fracturing experiments and the point where the two halves of the beam were separated and placed on a benchtop for the notched beam experiments. a Hydraulic fracturing experiment 2, where the point indicated by the asterisk (*) corresponds to the first event, subsequent tightening of the fitting resulting in a slight pressure decrease, and continued pressurization with a reduction of the injection rate from 1 to 0.2 ml/min. b Hydraulic fracturing experiment 3, which was performed in the same block and 20 mm above experiment 2. c Notched beam experiment 2

Figure 10 presents cumulative post unloading events versus the shifted time since unloading along with best-fit prediction based on frequency decay following the Omori law for the experiments not included in the main body of the paper. The observations are consistent with Fig. 7 in the main article, namely: (1) aftershocks for both types of experiments follow an Omori’s law frequency decay, (2) for the hydraulic fracturing experiments, we consistently obtain a change in slope at about 1000 s after shut-in and pressure relief.

Fig. 10

Cumulative post unloading events versus the shifted time since unloading along with best-fit prediction based on frequency decay following the Omori law for the experiments not included in the main body of the paper. For the hydraulic fracturing experiments the data have been fit with two lines, for $0<t<1000$ s and $t>1000$ s. a Hydraulic fracturing experiment 2. b Hydraulic fracturing experiment 3, which was performed in the same block and 20 mm above experiment 2. c Notched beam experiment 2, wherein it is apparent that the events tend to be temporally clustered

Appendix 2: Static fatigue laws

It is usually assumed (Scholz 1968) that the aftershocks are caused by local fractures (microcracks) of a particular type generated with time under subcritical stress—the stress below the failure stress. This phenomenon called static fatigue or delayed failure and is usually described by two types of laws: the exponential law and the power law.

The exponential law was developed by Zhurkov in 1965 [see reprint in Zhurkov (1984)] in the frame of his theory of kinetic fracture. It is assumed that the delayed failure is caused by thermal-fluctuation mechanism and the average life time of a sample loaded to stress is determined by the Arrhenius-type law

$$\begin{aligned} \tau =\tau _0\exp \left[ \frac{U_0-\gamma \sigma }{\hat{k}T}\right] \end{aligned}$$

(8)

where $\tau _0$, $U_0$ and $\gamma$ are material parameters: the characteristic time of elementary failure, the activation energy and the parameter characterizing the mechanical properties of the material, respectively (Kuksenko et al. 1996), $\hat{k}$ is Boltzman’s constant, and $T$ is the absolute temperature.

We now determine the probability of failure under the given stress. We note that the average time before failure referred to in Eq. (8) is the lifetime averaged over a number of independently loaded samples. In this case the average lifetime is $\tau =1/p_{\sigma }$, where $p_{\sigma }$ is the probability of sample failure under stress $\sigma$ during time $\tau _0$ (the probability rate). From here,

$$\begin{aligned} p_{\sigma }=\frac{1}{\tau _0}\exp \left[ -\frac{U_0-\gamma \sigma }{\hat{k}T}\right] \end{aligned}$$

(9)

Scholz (1968) used a similar form of static fatigue law in his analysis.

On the other hand, the power law has been found empirically for different types of loading and different brittle materials and rocks. It is usually refereed to in relation with the effect of environment such as moisture [see review by Gy (2003)]. It can be expressed as

$$\begin{aligned} \tau =\tau _0\left( \frac{\sigma }{\sigma _m}\right) ^{-n} \end{aligned}$$

(10)

Here $\tau _0$ is the characteristic time of elementary failure, $\sigma _m$ is the stress of microcrack initiation, and exponent $n$ is an experimentally-determined parameter. Similar to Eq. (9), the probability of sample failure under stress $\sigma$ during time $\tau _0$ (probability rate) using the power law of static fatigue is

$$\begin{aligned} p_{\sigma }=\frac{1}{\tau _0}\left( \frac{\sigma }{\sigma _m}\right) ^n \end{aligned}$$

(11)

The static fatigue behavior of the South Australian gabbro was tested using un-notched beams that were loaded in 3-point bending (Fig. 11a) up to a fixed load that was held until the specimen ruptured. The loading was ramped up over a period of 12 s. Figure 11b, c shows the empirical relationship between $\langle t \rangle$ and $\sigma$, where the stress $\sigma$ is given for this experimental configuration by $\sigma =3 P L/(2u d^2)$, where $P$ is the applied load, $L$ is the span between the two lower supports, and $d$ and $u$ are the specimen thicknesses measured in the direction parallel to and normal to loading, respectively. From these results it is clear that under tensile loading (produced in this case by bending), the relationship between the time to failure and the applied stress can be equally well-described as negative exponential (Fig. 11b) or power-law (Fig. 11b).

We conclude that in spite of the difference in mathematical form between the two probability rate laws embodied by Eqs. (9) and (11), existing experimental data is insufficient to judge which more accurately describes the behavior of rocks (Botvina (e.g. 1999) and Fig. 11). Following the route of Scholz (1968) one can obtain Omori’s law using the exponential static fatigue law (Eq. 9), while on the other hand, the analysis in the main paper shows that the power law relationship leads to Omori’s law when one makes a different assumption to Scholz (1968) regarding the relaxation behavior of the rock during microcrack generation.

Fig. 11

The static fatigue behavior of the South Australian gabbro, showing a the experimental setup, b empirical negative exponential relationship between $\langle t \rangle$ and $\sigma$, and c empirical power-law relationship between $\langle t \rangle$ and $\sigma$