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Self-Similar Solutions for a Fractional Thin Film Equation Governing Hydraulic Fractures

Abstract

In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient K ≥ 0. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness (K = 0) and the finite toughness K ∈ (0, ∞) cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all K > 0 there exists an injection rate for the fluid such that self-similar solutions exist.

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Correspondence to A. Mellet.

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Communicated by L. Caffarelli

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Imbert, C., Mellet, A. Self-Similar Solutions for a Fractional Thin Film Equation Governing Hydraulic Fractures. Commun. Math. Phys. 340, 1187–1229 (2015). https://doi.org/10.1007/s00220-015-2459-9

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Keywords

  • Stress Intensity Factor
  • Green Function
  • Hydraulic Fracture
  • Injection Rate
  • Free Boundary Condition