Realistic fracture simulations in rock as a heterogeneous brittle material with significant inherent randomness, require the use of models that incorporate its inhomogeneities and statistical variability. Since brittle materials do not match ductile materials in dissipating energy in the bulk, their fracture response is highly dependent on the stochastic microscale distribution and strength of defects. The high dependence of their fracture progress on microstructural defects results in wide scatter in their ultimate strength and the so-called size effect. Our approach for incorporating randomness in rocks is based on the modeling of stochastic volume elements (SVEs). Although representative volume elements (RVEs) are more commonly used in solid mechanics, SVEs are more appropriate for fracture analysis since they ensure that the material randomness is maintained. They still average microscale features similar to RVEs, and provide a more economical solution approach than those methods that explicitly model all microcracks in rock. To create a random field for macroscopic fracture strength field, we first generate several realizations of rock with a prescribed crack density and distribution. SVEs are then constructed with their centers at known spatial position on these random realizations. Next, by using a moving window approach, where the SVE traverses the known positions in these random realizations, we obtain first and second moments of the target random field. Point-wise probability distribution function and spatial covariance function are derived and used to generate consistent realizations of random fields based on the Karhunen-Loève (KL) method. Finally, such realizations will be used for the analysis of dynamic stimulation of a wellbore in a tight formation. A powerful and mesh adaptive spacetime discontinuous Galerkin finite element method is used for dynamic fracture simulations.

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