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arXiv · 1403.7491

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

Cyril Imbert·Antoine Mellet
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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 $K\geq 0$. 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 $K\textgreater{}0$ there exists an injection rate for the fluid such that self-similar solutions exist.

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Cyril Imbert, Antoine Mellet. 2016-01-03. Self-similar solutions for a fractional thin film equation governing hydraulic fractures. https://doi.org/10.1007/s00220-015-2459-9

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