Abstract
The work presented herein aims at characterizing and modeling fracturing (i.e., initiation and propagation of cracks) in a clay-rich rock. The analysis is based on two experimental campaigns. The first one relies on a probabilistic analysis of crack initiation considering Brazilian and three-point flexural tests. The second one involves digital image correlation to characterize crack propagation. A nonlocal damage model based on stress regularization is used for the simulations. Two thresholds both based on regularized stress fields are considered. They are determined from the experimental campaigns performed on Lower Watrous rock. The results obtained with the proposed approach are favorably compared with the experimental results.
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Abbreviations
- \({\varvec{\sigma }}\) :
-
Stress tensor
- \(\overline{{\varvec{\sigma }}}\) :
-
Regularized stress tensor
- \(\ell _{\mathrm{c}}\) :
-
Characteristic length
- \({\varDelta }\) :
-
Laplacian operator
- \({\varvec{n}}\) :
-
Normal to surface
- \(P_{\mathrm{i}}\) :
-
Crack initiation probability
- \(S_{\mathrm{i}}\) :
-
Crack initiation stress
- \(\frac{\sigma _{0}^{m}}{\lambda _{0}}\) :
-
Scale parameter
- \(V_{\mathrm{el}}\) :
-
Volume of an element
- m :
-
Weibull modulus
- \(\sigma _{\ell _{\mathrm{c}}}\) :
-
Nominal stress
- \(S_{\mathrm{g}}\) :
-
Crack growth stress
- \(K_{\mathrm{c}}\) :
-
Fracture toughness
- \({\varGamma }\) :
-
Gamma function
- \(\rho\) :
-
Mass density
- d :
-
Damage
- \(\psi _{\mathrm{e}}\) :
-
State potential
- \({\mathcal {C}}\) :
-
Hooke’s tensor
- \({\varvec{\epsilon }}\) :
-
Infinitesimal strain tensor
- Y :
-
Thermodynamic force associated with damage
- \(H_{\mathrm{e}}\) :
-
Heaviside function
- \({\overline{\sigma }}_{\mathrm{I}}\) :
-
Maximum principal regularized stress
- a :
-
Half-length of critical defect
- \(\sigma _{\mathrm{w}}\) :
-
Weibull stress
- \(\sigma _{\mathrm{w}i}\) :
-
Weibull stress associated with sample i
- \(P_{\mathrm{F}}\) :
-
Failure probability
- \(P_{\mathrm{F}i}\) :
-
Failure probability associated with sample i
- \(\sigma_{\mathrm{f}}\) :
-
Critical maximum principal stress
- H :
-
Stress heterogeneity factor
- V :
-
Sample volume
- \(H_{\mathrm{br}}\) :
-
Stress heterogeneity factor for Brazilian test
- \(V_{\mathrm{br}}\) :
-
Sample volume for Brazilian test
- \(H_{\mathrm{fl}}\) :
-
Stress heterogeneity factor for three-point flexural test
- \(V_{\mathrm{fl}}\) :
-
Sample volume for three-point flexural test
- \(n_{\mathrm{s}}\) :
-
Total number of samples
- R :
-
Radius of Brazilian test sample
- L :
-
Length of Brazilian test sample
- \(K_{\mathrm{I}}\) :
-
Mode I stress intensity factor
- \(K_{\mathrm{II}}\) :
-
Mode II stress intensity factor
- \(r_{\mathrm{K}}\) :
-
Ratio between mode II and mode I stress intensity factors
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Acknowledgements
This work was funded by BRGM through an “Institut Carnot” research Grant. The authors wish to thank Dr. Steve Whittaker and Saskatchewan Industry and Resource for kindly providing the samples of Lower Watrous caprock.
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Guy, N., Seyedi, D.M. & Hild, F. Characterizing Fracturing of Clay-Rich Lower Watrous Rock: From Laboratory Experiments to Nonlocal Damage-Based Simulations. Rock Mech Rock Eng 51, 1777–1787 (2018). https://doi.org/10.1007/s00603-018-1432-2
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DOI: https://doi.org/10.1007/s00603-018-1432-2