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Communications in Mathematical Physics

, Volume 340, Issue 3, pp 1187–1229 | Cite as

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

  • C. Imbert
  • A. MelletEmail author
Article

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.

Keywords

Stress Intensity Factor Green Function Hydraulic Fracture Injection Rate Free Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CNRS, UMR 7580Université Paris-Est CréteilCréteil CedexFrance
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Fondation Sciences Mathématiques de ParisParis Cedex 05France

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