Energy Budget of Brittle Fracturing in Granite Under Stress Relaxation and Creep

Abstract

Creep and relaxation are the two major time-dependent fracturing processes in rocks. While a considerable amount of research has been done in understanding these two mechanisms, critical gaps remain regarding how different energy components evolve during time-dependent fracturing processes in rocks. In this study, a series of relaxation and creep experiments were conducted on prismatic Barre granite specimens in the laboratory to estimate the energy budget of brittle fracturing in granite. For the input energy, the work done by the machine (W) is calculated and for the output energy the radiated seismic energy (${E}_{R})$, released in the form of acoustic emissions (AEs), is calculated as the only measurable output energy component in the conducted experiments. The low-frequency plateau (${\Omega }_{0})$ and corner frequency (${f}_{0})$ for each AE waveform was estimated by fitting the observed AE spectra with the theoretical spectra using the Omega model. These parameters were used to estimate the seismic moments (${M}_{0})$ based on the radiation pattern for the double couple (shear) and non-double-couple (non-shear) events. The range of ${f}_{0}$ and ${M}_{0}$ varied from 150 to 750 kHz and 10⁻⁴ to 10⁻¹ N m, respectively. Moment magnitude (${M}_{w})$ varied in a wider range from − 9 to − 6 in creep and − 8.5 to − 7 in relaxation. Stress drops ($\Delta \sigma )$ and source radius ($r)$ were estimated for the AEs using Brune’s model. The results report on three primary observations: (1) the effects of different source mechanisms on the estimated source parameters showed that ${M}_{0}$ and $\Delta \sigma$ were higher for DC events as compared to NDC in both relaxation and creep. (2) The radiation efficiency in the case of creep is 70% higher as compared to relaxation and, (3) the stress drop estimated in relaxation and creep demonstrated a breakdown in scaling with the seismic moment.

Highlights

• Investigating the impact of various fracture mechanisms (e.g. tensile, shear) on the energy budget.

• Effect of time-dependent behavior (e.g., creep, relaxation) on energy budget

• Non-self-similar behavior between stress drop and seismic moment was observed for the AEs produced under creep and relaxation

Appendices

Appendix A: Absolute Calibration of AE Sensors

Calibration of the AE sensors is important for the estimation of the reliable source parameters behind the generation of an AE signal and to investigate the scaling relationship between the laboratory AEs and field seismicity. In the past, various experimental techniques have been developed to calibrate the AE sensors based on the measurements from a known source (glass capillary fracture or ball drop) (McLaskey and Glaser 2010, 2012). The methods detailed by (McLaskey and Glaser 2012) were adopted to calibrate the Nano 30 sensors. The performance of Nano 30 sensors was characterized by an elastic wave propagation through an elastic isotropic, homogeneous steel transfer plate generated by a series of ball drop experiments. The theory related to the calibration of the AE sensors using ball drop is well described in the literature (Selvadurai 2019; Wu et al. 2021). The goal of the sensor calibration procedure is to relate the theoretical disturbance at the sensor location for a known source.

Figure 15 shows the force–time function at the source due to the ball impact and the corresponding velocity signal at the sensor location in the time domain. The source function (Fig. 15) generated by the ball impacts were calculated based on Hertzian theory (as reported in McLaskey and Glaser 2010), through the following equation:

$$f\left(t\right)=\left\{\begin{array}{ll}{f}_{\mathrm{max}}\sin({\frac{\pi t}{{t}_{c}})}^\frac{3}{2} \quad 0\le \left|t\right| \le {t}_{c}\\ 0 \qquad \qquad \qquad {\rm otherwise},\end{array}\right.$$

(10)

where ${t}_{c}=4.53 ({\frac{4{\rho }_{1}\uppi \left({\delta }_{1 }+ {\delta }_{2}\right)}{3})}^{2/5}{R}_{1 }{{v}_{0}}^{-\frac{1}{5}}$ is the contact time the ball spends with the steel plate. The maximum force is calculated by the equation ${f}_{\mathrm{max}}=1.917 {\rho }_{1}^\frac{3}{5}{({\delta }_{1}+{\delta }_{2})}^{-2/5}{R}_{1}^{2}{{v}_{0}}^\frac{6}{5}$. In these equations, ${\delta }_{i}=(1-{\vartheta }_{i}^{2})/\pi {E}_{i}$, and $E$ and $\vartheta$ are the Young’s modulus and Poisson’s ratio, respectively. ${R}_{1}$ is the radius and ${v}_{0}$ is the incoming velocity of the ball. Subscript 1 refers to the properties of the material of the ball and subscript 2 refers to the material of the steel plate.

Fig. 15

Calibration formulation to obtain the instrument response for Nano 30 against velocity (Zafar et al. 2022c). a Source function for ball drop convolved (⦻) with the green’s function ($g(t))$ gives the b theoretical velocity ($u (x,t))$ at the sensor position; c Fourier transform of $u (x,t)$ and (d) Fourier transform of the recorded signal $s (x,t)$. Instrument response $I(\omega )$ is computed by dividing $s \left(x,\omega \right)$ with $u \left(x,\omega \right)$ as shown in Eq. (2)

AE signals generated through the impact force were acquired by connecting a DAQ system with a sampling frequency of 5 MHz. The gain in the pre-amplifiers was set as 20 dB and the detection threshold was set as 55 dB. These settings were identical to those used in the fracturing experiments.

The instrument response function can be calculated by dividing the recorded signal (Fig. 15d) with the theoretically calculated signal in the frequency domain (Fig. 15c), as shown by the following equation (McLaskey and Glaser 2012)

$$I\left(\omega \right)=S(x,\omega )/U(x,\omega )$$

(11)

where $S(x,\omega )$ is the recorded signal in the frequency domain and $U(x,\omega )$ is the theoretical disturbance (displacement, velocity or acceleration) obtained through the convolution of the force function and the green’s function at the sensor position.

The acquired waveforms were windowed with a Blackman Harris window (as reported in Wu et al. 2021) for a window length of 56 μs centered about the P-wave arrival. The frequency analysis was carried out for the windowed signal (Selvadurai 2019; Wu et al. 2021). Figure 16a shows one of the AE signal recorded in the relaxation experiment, its FFT and spectrogram in the time–frequency domain. Similarly, Fig. 16b illustrates the arrival portion of the waveform considered for the sensor calibration, its FFT and spectogram in the time–frequency domain. In this study, the frequency spectra of the waveforms obtained from the sensors indicated that these sensors produced a signal proportional to velocity. Hence, the velocity instrument responses obtained through the sensor calibration for each sensor were used to convert the FFTs of the AEs generated in the fracturing experiments to their corresponding velocity spectra. The velocity spectra obtained from the deconvolution were further intergrated to get the displacement spectra. The displacement spectra were smoothed by using an eight-point average moving filter. The noise floor spectra were obtained for 56 μs before the P-wave arrival (Fig. 17).

Fig. 16

a An AE signal detected by sensor 4 with FFT of the waveform in the frequency domain and its spectrogram in the time–frequency domain; b arrival portion of the waveform shown in the inset considered for sensor calibration, its FFT and spectrogram in the time–frequency domain

The displacement spectrum is then fitted by Omega model (as reported in Goodfellow et al. 2015) to determine the low-frequency plateau (Ω₀) and the corner frequency (${f}_{0}):$

$$\Omega \left(f\right)=\frac{{\Omega }_{o}{e}^{(-\frac{\pi fR}{VQ})}}{1+({\frac{f}{{f}_{0}})}^{n}}$$

(12)

where $R$ is the source-receiver distance, $V$ is the P-wave velocity, n is the frequency fall-off rate and $Q$ is the frequency-independent attenuation coefficient, taken as 38.39 for granite (Li and Einstein 2020). The parameters ${\Omega }_{0}$ and ${f}_{0}$ were obtained by fitting the Omega model to the displacement spectra of the individual waveforms using a non-linear least square method in the frequency range of 100 kHz to 1 MHz, as shown in Fig. 17.

Fig. 17

Fourier displacement spectrum for one of the waveforms and the fitted Omega-model as per Eq. (12). ${\Omega }_{0}$ and ${f}_{0}$ obtained from the fitting is used to evaluate the source parameters (Zafar et al. 2022a, b, c)

Appendix B

The material in this study was treated as isotropic and the P-wave velocities measured in the horizontal and vertical directions were approximately 4000 and 4200 m/s, respectively. As discussed in Sect. 4.1.1, during the fracturing experiment, the P-wave velocity in Barre granite showed a variation of ± 5% (ranging from 3895 to 4305 m/s) with time. A variation in the P-wave velocity during the fracturing experiment can introduce errors in the estimated AE locations as the damage progresses. To quantify the errors in the determination of the AE source location due to the 5% variation in the P-wave velocity, a sensitivity analysis was done using a ball drop test.

Ball drop experiment to quantify the errors in the AE source location using different velocity models:

A 0.3 mm steel ball was dropped from a fixed height at five known locations on the rock specimen and the signals were recorded by 12 AE sensors. The location of the AE sensors and the AE settings of the data acquisition system were the same as the relaxation and creep experiments with the threshold 55 dB, gain 20 dB, and sampling rate 5 MHz.

Table

Table 3 Difference between the estimated AE locations using different velocity models and the actual source location for Ball drop at 5 known locations along with the error:

3 shows the variations in the estimated source location by AE with actual ball drop location using different P-wave velocities, i.e., v = 3895 m/s, v = 4100 m/s, and v = 4305 m/s. The error in mm was estimated by calculating the distance between the predicted AE locations (X₀ and Y₀) and the actual AE coordinates (X and Y) for different P- wave velocities. The error varied in the range of 0.5–1.9 mm and fell below the selected error tolerance (± 2 mm) for this study; hence, the material was treated as isotropic.