Improving the Accuracy of Fracture Toughness Measurement in Burst Experiments

Abstract

Experimental studies suggest that the fracture toughness of rocks increases with the confining pressure. Among many methods to quantify this dependency, a so-called burst experiment (Abou-Sayed, 1978) may be the most widely applied in practice. Its thick wall cylinder geometry leads to a stress state resembling the subsurface condition of a pressurized wellbore with bi-wing fractures. The fracture toughness of a sample, under a given confinement pressure, can be recovered from the critical pressure upon which the bi-wing cracks propagate. Traditionally, this critical pressure is thought to correspond to a sudden drop in injection pressure. However, as the standard configuration was deliberately designed to obtain stable fracture growth at the onset, propagation can take place well before this drop in pressure, and one may overestimate the fracture toughness from measured pressures. Here, we study crack stability in the burst experiment and propose modifications to the experimental design which promotes unstable fracture growth and makes the critical pressure less ambiguous to interpret. We found that experiments with the original, stable design can lead to inconsistent measurement of fracture toughness under confining pressure, while results from unstable configurations are more consistent. Our claim on the stability was also supported by the recorded acoustic emissions from both stable and unstable experiments.

Highlights

• We analyzed stability of thick wall cylinder experiments for fracture toughness estimation under confining pressure known as the burst experpiment.

• Our analysis demonstrated that fracture toughness may systematically be overestimated in the original burst experiment configuration.

• We propose modifications to the burst experiment for a more accurate fracture toughness measurement.

• Our experiments confirmed the superiority of the modified design for detecting the onset of fracture propagation and measuring the fracture toughness of a sample.

Appendix 1: Stress intensity factor computation

In the original burst experiment proposal, Abou-Sayed (1978) decomposed the problem into the jacketed (J) and the unjacketed problems (U) (Fig. 8) as

$$\begin{aligned} {\widetilde{K}}_\mathrm {I}(w,\ell ,p^*) \approx {\widetilde{K}}_\mathrm {I}^J(w,\ell ) - p^* {\widetilde{K}}_\mathrm {I}^U(w,\ell ), \end{aligned}$$

(A1)

and for $K_\mathrm {I}^{J*}$ and $K_\mathrm {I}^{U*}$ computation, the results from Clifton et al. (1976) were used.

Fig. 8

Approximated superposition of the burst SIF proposed in Abou-Sayed (1978)

Note that this decomposition is not exact, because the unjacketed stress intensity factor is not equivalent to the constrained stress intensity factor ($K_\mathrm {I}^{U*} \ne K_\mathrm {I}^{C*}$). Here, we computed all the stress intensity factors numerically using the $G-{\theta }$ method (Dubois et al. 1998). Figure 9 compares $K_\mathrm {I}^{J*}$ and $K_\mathrm {I}^{U*}$ from the $G-{\theta }$ method with the reported values from Clifton et al. (1976) and the results are in close agreement.

Fig. 9

Comparison of computed SIF evolutions with $\ell$ with Clifton et al. (1976)

Fig. 10

Normalized error caused by approximating $K_{I}^{U*}$ with $K_{I}^{C*}$

Because the difference between Eq. (9) and Eq. (A1) amounts to the difference between the constrained and the unjacketed stress intensity factors ($K_\mathrm {I}^{C*}$ and $K_\mathrm {I}^{U*}$), we computed the relative errors ($(K_\mathrm {I}^{U*} - K_\mathrm {I}^{C*})/K_\mathrm {I}^{U*}$ ) with various wall thicknesses (w) against fracture lengths in Fig. 10. The approximation in Eq. (A1) may be acceptable for long fractures ($\ell >0.8$), but the errors increase for short fractures.