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


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.


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