## Abstract

The paper deals with the plane strain problem of a shallow fluid-driven fracture propagating parallel to a free surface in an impermeable elastic rock. For a shallow fracture, the aperture corresponds essentially to the deflection of the layer defined by the free-surface and the crack. This problem is thus governed by the Euler-Bernoulli beam equation and by the lubrication theory. We show that the solution generally depends on two independent time scales and that there exist 4 similarity solutions denoted as Ō, S̄, M̄ and K̄. Furthermore, we demonstrate that the parametric space of solution is a rectangle with the vertices corresponding to the similarity solutions. The Ō- and S̄-solutions are two different early time solutions characterized by a small length of the fluid-infiltrated region compared to the fracture length (i.e., the fluid lag occupies nearly the whole fracture), while the M̄ and A solutions are two large-time solutions characterized by a vanishingly small lag. The M̄- and the K̄- vertices are similarity solutions for the particular case when Gc = 0, while Ō- and S̄- are similarity solutions for the case σo = 0. Thus the S̄M̄-edge and the S̄M̄-edge represent two particular fracture evolutions, the first one when Gc = 0 and the second one when σo = 0. However, the large time solution for the general case Gc > 0 and σo > 0 has not yet been identified, as the numerical solutions conflict with the large time matched asymptotic solution.

Original language | English (US) |
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Title of host publication | 57th US Rock Mechanics/Geomechanics Symposium |

Publisher | American Rock Mechanics Association (ARMA) |

ISBN (Electronic) | 9780979497582 |

DOIs | |

State | Published - 2023 |

Externally published | Yes |

Event | 57th US Rock Mechanics/Geomechanics Symposium - Atlanta, United States Duration: Jun 25 2023 → Jun 28 2023 |

### Publication series

Name | 57th US Rock Mechanics/Geomechanics Symposium |
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### Conference

Conference | 57th US Rock Mechanics/Geomechanics Symposium |
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Country/Territory | United States |

City | Atlanta |

Period | 6/25/23 → 6/28/23 |

### Bibliographical note

Publisher Copyright:© 2023 57th US Rock Mechanics/Geomechanics Symposium. All Rights Reserved.