Self-similar solutions for a fractional thin film equation governing hydraulic fractures

Cyril Imbert (1) , Antoine Mellet (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Department of Mathematics [College Park]

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 $K\geq 0$. 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 $K\in(0,\infty)$ 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.

Mots clés: Hydraulic fractures; higher order equation; thin films; fractional Laplacian; self-similar solutions; free boundary condition

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Antoine Mellet. Self-similar solutions for a fractional thin film equation governing hydraulic fractures. Communications in Mathematical Physics, 2015, 340 (3), pp.1187-1229 ⟨10.1007/s00220-015-2459-9⟩. ⟨hal-00967393v2⟩