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A variational phase field model of hydraulic fracturing Tanne, Erwan


Since their inception in the mid-90's, variational phase-field models of fracture [1] have steadily gained popularity. One of their strengths, the ability to handle complex topologies with unknown crack paths and the interaction between multiple cracks, which is a fundamental requirement for the numerical simulation of hydraulic fracturing in complex situations. Following the technique of [2], crack propagation subject to given pressure $p$ acting along the fracture surfaces of a brittle material occupying a domain $\Omega$ is computed as the minimizer of the energy functional $$ \mathcal{E}_\ell (u,\alpha)= \int_{\Omega} \frac{1}{2} (1-\alpha)^2 \mathtt{A} e(u):e(u) \mathrm{d}x - p \int_{\Omega} \alpha \div(u) \mathrm{d}x + \frac{3G_c}{8} \int_{\Omega} \frac{\alpha}{\ell} + \ell | \nabla \alpha |^2 \mathrm{d}x $$ where $u$ denotes the displacement field, $\alpha$ the phase field representing the fracture geometry and $\mathtt{A}$ its Hooke's law. As in [2], in the case of a single pre-existing line or penny-shaped crack in an infinite medium the pressure and volume of fluid recovered from minimizers of (1) can be compared with solutions from the literature [3]. This formalism can also be used to address the issue of hydraulic stimulation of multiple cracks. Symmetry arguments are routinely used to suggest that the propagation of an infinite array of cracks of equal length in an infinite reservoir is possible. Yet a simple stability analysis reveals that this is not the case and that loss of symmetry is always energetically favored [4]. References [1] Bourdin, B., Francfort, G., and Marigo, J.-J., Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids, (2000), 48(4) 797-826 [2] Bourdin, B., Chukwudozie, C., and Yoshioka, K. (2012). A variational approach to the numerical simulation of hydraulic fracturing. In Proceedings of the 2012 SPE Annual Technical Conference and Exhibition, volume SPE 159154. [3] Sneddon, I. and Lowengrub, M. (1969). Crack problems in the classical theory of elasticity. The SIAM series in Applied Mathematics. John Wiley & Sons. [4] Tanné, E. (2017). Variational phase-field models from brittle to ductile fracture: nucleation and propagation. PhD thesis, Université Paris-Saclay, Ecole Polytechnique.

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